Basics of Electric Fields
| Sitio: | Young Education |
| Curso: | Electric and Magnetic Fields |
| Libro: | Basics of Electric Fields |
| Impreso por: | Guest user |
| Fecha: | lunes, 9 de noviembre de 2026, 16:38 |
1. Electric charge
Learning outcomes
- I can describe electric charge as a fundamental property of matter.
- I can distinguish between positive and negative electric charges.
- I can explain how like charges repel and opposite charges attract.
- I can describe how electric charge is transferred through friction, conduction, and induction.
- I can explain the principle of conservation of charge in physical systems.
Introduction
Have you ever rubbed a balloon on your hair and watched it stick to a wall? Or noticed a small shock after walking across a carpet and touching a metal door handle? These everyday experiences are caused by electric charge.
Electric charge is one of the most fundamental properties of matter. It is responsible for electric forces, electric fields, lightning, and many of the technologies we use every day. Understanding electric charge is the first step toward learning about electric fields, electric circuits, magnetism, and modern electronics.
What Is Electric Charge?
Electric charge is a fundamental property of matter that causes objects to experience electric forces.
Every atom contains charged particles:
- Protons, which have a positive (+) charge.
- Electrons, which have a negative (–) charge.
- Neutrons, which have no charge.
An object's overall charge depends on the balance between its protons and electrons.
Definition:
Electric charge is a fundamental property of matter that causes objects to attract or repel one another through electric forces.
Positive and Negative Charge
There are two types of electric charge:
- Positive (+)
- Negative (–)
Normally:
- Protons remain fixed inside the nucleus.
- Electrons can move from one object to another.
An object becomes:
- positively charged if it loses electrons,
- negatively charged if it gains electrons,
- electrically neutral if it has equal numbers of protons and electrons.
| Object | Charge |
|---|---|
| More protons than electrons | Positive |
| More electrons than protons | Negative |
| Equal protons and electrons | Neutral |
Worked Example 1
A neutral atom loses two electrons.
Question
What is the overall charge of the atom?
Solution
The atom now has two more protons than electrons, so it has a +2 charge.
How Charged Objects Interact
Charged objects exert forces on one another.
The rules are simple:
- Like charges repel.
- Unlike (opposite) charges attract.
This can be summarised as:
| Charges | Force |
|---|---|
| Positive and Positive | Repel |
| Negative and Negative | Repel |
| Positive and Negative | Attract |
These forces act even when the objects are not touching.
Why Do Charges Attract and Repel?
Charged particles produce electric fields around themselves.
When another charged particle enters this field, it experiences an electric force.
For example:
- Two negatively charged balloons push apart.
- A positively charged object attracts a negatively charged object.
- Lightning occurs because enormous electric charges build up in clouds and then discharge.
The strength of the force depends on:
- the amount of charge,
- the distance between the charges.
(You will study this in more detail when learning Coulomb's Law.)
Transferring Electric Charge
Electric charge is usually transferred by the movement of electrons, not protons.
There are three main methods of charging an object:
- Friction
- Conduction
- Induction
Charging by Friction
When two different materials are rubbed together, electrons may move from one material to the other.
One object:
- loses electrons and becomes positively charged.
The other:
- gains electrons and becomes negatively charged.
Example
Rubbing a balloon on your hair:
- electrons move from your hair to the balloon,
- the balloon becomes negatively charged,
- your hair becomes positively charged.
The opposite charges attract, causing your hair to stand up or the balloon to stick to a wall.
Charging by Conduction
Conduction occurs when a charged object touches another object.
Electrons flow between the objects until the charges are redistributed.
Example:
- A negatively charged metal rod touches a neutral metal sphere.
- Some electrons move onto the sphere.
- Both objects become negatively charged.
Conduction requires physical contact.
Charging by Induction
Induction allows an object to become charged without touching the charged object.
A nearby charged object causes electrons inside a conductor to move.
If the conductor is grounded while the charged object is nearby, electrons can enter or leave the conductor. Removing the ground and then the nearby charged object leaves the conductor with a net charge.
Induction is widely used in:
- electrostatic painting,
- photocopiers,
- lightning protection.
Comparing the Three Methods of Charging
| Method | Contact Required? | How Charge Is Transferred |
|---|---|---|
| Friction | ✔ Yes | Electrons move during rubbing |
| Conduction | ✔ Yes | Electrons move through direct contact |
| Induction | ✘ No | Charges rearrange due to a nearby charged object, often with grounding |
Conservation of Charge
One of the most important laws in physics is the Law of Conservation of Charge.
It states:
Electric charge cannot be created or destroyed. It can only be transferred from one object to another.
When one object gains electrons:
- another object loses the same number of electrons.
The total electric charge in a closed system remains constant.
Worked Example 2
A neutral metal sphere gains 5 electrons.
Question
What is its new charge?
Solution
Each electron carries one negative elementary charge.
The sphere now has 5 extra electrons, so its overall charge is –5e, where e is the elementary charge.
Worked Example 3
A student says:
"When an object becomes positively charged, it has gained protons."
Question
Is this correct?
Solution
No.
In ordinary charging processes, protons remain in the nucleus.
An object becomes positively charged because it loses electrons.
Everyday Examples of Electric Charge
Electric charge is responsible for many familiar phenomena.
Examples include:
- lightning,
- static shocks after walking on carpet,
- clothes sticking together after drying,
- balloons sticking to walls,
- dust sticking to television screens,
- electrostatic air filters,
- photocopiers and laser printers.
Understanding electric charge helps explain these everyday observations.
Real-World Connection
Electrostatic charging is used in many industries. Spray painters use charged paint droplets so that paint is attracted evenly to metal surfaces, reducing waste. Air purifiers and electrostatic precipitators remove tiny dust particles from the air by charging them and collecting them on oppositely charged plates. Engineers also design electronic devices to prevent damage from static electricity, which can destroy delicate microchips.
Did You Know?
A single lightning bolt can transfer tens of coulombs of charge and carry currents of tens of thousands of amperes. The lightning channel can reach temperatures of about 30,000°C, which is roughly five times hotter than the surface of the Sun!
Key Terms
- Electric charge — a fundamental property of matter that causes electric forces.
- Positive charge — the type of charge associated with protons or an object that has lost electrons.
- Negative charge — the type of charge associated with electrons or an object that has gained electrons.
- Neutral object — an object with equal numbers of protons and electrons.
- Friction — a method of charging by rubbing materials together.
- Conduction — charging by direct contact, allowing electrons to move between objects.
- Induction — charging without direct contact by causing electrons to redistribute.
- Conservation of charge — the principle that electric charge cannot be created or destroyed, only transferred.
- Electric field — the region around a charged object where another charge experiences an electric force.
Key Takeaways
- Electric charge is a fundamental property of matter arising from the presence of protons and electrons.
- There are two types of charge: positive and negative.
- Like charges repel, while opposite charges attract.
- Objects become charged through the movement of electrons, not protons.
- Electric charge can be transferred by friction, conduction, and induction.
- The Law of Conservation of Charge states that total electric charge remains constant and can only be transferred between objects.
- Understanding electric charge provides the foundation for studying electric fields, electric forces, circuits, and magnetism.
2. Coulomb’s Law
Learning outcomes
- I can describe Coulomb's Law and explain how electric force depends on charge and distance.
- I can use Coulomb's Law to calculate the electric force between two point charges.
- I can determine whether the electric force between charged objects is attractive or repulsive.
- I can analyze how changing charge magnitude or separation distance affects electric force.
- I can determine the net electric force on a charge in a system containing three or more charges using vector addition and the principle of superposition.
Electric Forces Between Charges
Electric charges exert forces on one another. These forces can act even when the charged objects are not touching, so electric force is a non-contact force.
There are two types of electric charge:
- Positive charge
- Negative charge
The basic interaction rule is:
Like charges repel.
Opposite charges attract.
Therefore:
- Positive + positive → repel
- Negative + negative → repel
- Positive + negative → attract
The strength of the electric force depends mainly on:
- The magnitude of the charges
- The distance between the charges
Coulomb's Law describes this relationship mathematically.
Coulomb's Law
For two point charges, the magnitude of the electric force is:
where:
- = magnitude of electric force in newtons (N)
- = first charge in coulombs (C)
- = second charge in coulombs (C)
- = distance between the centres of the charges in metres (m)
- = Coulomb constant
The Coulomb constant is approximately
.
The absolute-value signs around
mean that we calculate the magnitude of the force as a positive value. We then use the signs of the charges to determine whether the force is attractive or repulsive.
Point Charges
Coulomb's Law is written for point charges.
A point charge is an idealized object in which all the charge is treated as though it were concentrated at one location.
Real charged objects have physical dimensions, but the point-charge model can often be used when:
- The objects are very small compared with their separation.
- The charge distribution is sufficiently simple.
- Spherical charge distributions can be treated as though their charge were concentrated at their centres.
For spherical objects, the separation
is normally measured from centre to centre, not from surface to surface.
Charge and the Coulomb
Electric charge is measured in coulombs (C).
One coulomb is a large amount of charge compared with the charges commonly used in electrostatic calculations. Charges are therefore often given in:
- Microcoulombs ()
- Nanocoulombs ()
Remember that
.
Also,
.
Charges should normally be converted to coulombs before being substituted into Coulomb's Law.
Worked Example: Converting Charge
Convert
into coulombs.
Since
:
Similarly:
Calculating Electric Force
Suppose two point charges have values
and
.
They are separated by
.
Start with Coulomb's Law:
Substitute the values:
First multiply the charges:
Square the distance:
Therefore:
This gives:
Rounded appropriately, the electric force is approximately 0.22 N.
Because the charges have opposite signs, the force is attractive.
Attractive and Repulsive Forces
The signs of the charges determine the direction of the interaction.
For two positive charges:
Positive ← → Positive
They repel.
For two negative charges:
Negative ← → Negative
They also repel.
For opposite charges:
Positive → ← Negative
They attract.
The electric force acts along the imaginary straight line joining the two charges.
Newton's Third Law and Electric Forces
If Charge A exerts an electric force on Charge B, then Charge B exerts an equal-magnitude force in the opposite direction on Charge A.
In magnitude:
This is an example of Newton's Third Law.
Even if one charge is much larger than the other, the forces they exert on each other have equal magnitudes.
Their accelerations may be different if their masses are different.
How Charge Magnitude Affects Force
Coulomb's Law shows:
This means that electric force is directly proportional to the product of the charge magnitudes.
If one charge doubles:
Charge × 2 → Force × 2
If both charges double:
Charge 1 × 2 and Charge 2 × 2 → Force × 4
If one charge triples:
Charge × 3 → Force × 3
If one charge is halved:
Charge × 0.5 → Force × 0.5
Worked Example
Two charges initially exert a force of 6.0 N.
If one charge is doubled:
If both charges are doubled:
How Distance Affects Electric Force
Distance has an especially strong effect because it is squared in Coulomb's Law.
This is called an inverse-square relationship.
If the distance doubles:
Therefore:
Distance × 2 → Force × 1/4
Similarly:
- Distance × 3 → Force × 1/9
- Distance × 4 → Force × 1/16
- Distance × 1/2 → Force × 4
Relatively small changes in separation can therefore produce large changes in electric force.
Worked Example: Changing Distance
Two charges initially exert a force of 12 N.
Their separation is doubled.
If their separation were instead reduced to half:
Combining Changes in Charge and Distance
Sometimes both charge and distance change.
Suppose:
- Both charge magnitudes double.
- The distance doubles.
Doubling both charges changes the force by a factor of:
Doubling the distance changes the force by:
Combining the effects:
Therefore, the electric force remains unchanged.
Worked Example
Two charges initially exert a force of 8.0 N.
One charge is tripled and the distance is doubled.
Tripling the charge gives a factor of 3.
Doubling the distance gives a factor of
.
Therefore:
Electric Force Is a Vector
Electric force is a vector quantity.
A vector has:
- Magnitude
- Direction
With only two charges, determining the direction is usually straightforward.
With three or more charges, a charge may experience several electric forces simultaneously.
These forces must be combined using vector addition.
The Principle of Superposition
The principle of superposition states that the net electric force on a charge is the vector sum of the individual electric forces exerted by all the other charges.
In symbols:
A useful procedure is:
- Select the charge whose net force you want to find.
- Consider each other charge separately.
- Calculate the force produced by each charge.
- Determine the direction of each force.
- Add the forces as vectors.
Three Charges in a Straight Line
Consider three positive charges:
A (+) —— B (+) —— C (+)
We want to determine the net force on B.
Charge A repels B toward the right.
Charge C repels B toward the left.
Suppose:
Force from A on B = 5.0 N right
Force from C on B = 3.0 N left
Choose right as positive.
Therefore, the net force is 2.0 N to the right.
Worked Example: Attraction and Repulsion Together
Consider:
A (+) —— B (+) —— C (−)
We want the net force on B.
A and B are both positive, so they repel. Since A is to the left of B, A pushes B to the right.
Suppose this force is 4.0 N right.
B is positive and C is negative, so they attract. Since C is to the right of B, C pulls B to the right.
Suppose this force is 6.0 N right.
Both forces point in the same direction.
The net force is 10.0 N to the right.
Equal Forces Can Cancel
Consider:
A (+) —— B (+) —— C (+)
Suppose A and C have equal charges and are equal distances from B.
A repels B to the right.
C repels B to the left.
If each force has a magnitude of 4.0 N:
The net force is 0 N.
This does not mean that no electric forces are acting. Two forces are present, but they have equal magnitudes and opposite directions, so they cancel.
Two-Dimensional Charge Arrangements
Charges do not always lie along a straight line.
Suppose a charge experiences:
- 3.0 N to the right
- 4.0 N upward
The forces are perpendicular.
Use Pythagoras:
Substitute:
So:
The resultant force has a magnitude of 5.0 N.
To find its direction:
Therefore:
The net force is approximately 5.0 N at 53° above the horizontal.
A Strategy for Multi-Charge Problems
Identify the Target Charge
Determine which charge you are finding the net force on.
Consider One Other Charge at a Time
Temporarily ignore the remaining charges.
Ask whether each interaction is attractive or repulsive.
Determine the Direction First
Write or sketch the direction of each force before performing calculations.
Calculate Each Force
Use Coulomb's Law separately for each pair.
Add the Forces as Vectors
For forces along one line, choose a positive direction and use positive and negative values.
For two-dimensional situations, resolve forces into horizontal and vertical components.
Superposition in Two Dimensions
Add all horizontal components:
Add all vertical components:
Then calculate the resultant magnitude:
The direction can be calculated using:
The signs of the components must also be considered to determine the correct direction or quadrant.
Coulomb's Law and Gravity
Coulomb's Law has an interesting similarity to Newton's Law of Universal Gravitation.
Both are inverse-square laws.
Electric force:
Gravitational force:
In both cases, increasing the distance greatly decreases the force.
However, there is an important difference:
Gravity between ordinary masses is always attractive.
Electric forces can be:
- Attractive
- Repulsive
depending on the signs of the charges.
Real-World Importance of Electric Forces
Electric forces are responsible for many familiar and microscopic effects.
They play important roles in:
- Static electricity
- Interactions between electrons and atomic nuclei
- Chemical bonding
- Photocopiers and laser printers
- Electrostatic painting
- Electrostatic precipitators
- Electronic devices
At the atomic level, electric forces are fundamental to the structure and behaviour of matter.
Common Mistakes
Forgetting to Convert Microcoulombs
If
, do not substitute 4.0 directly into Coulomb's Law.
Convert first:
Forgetting to Square the Distance
Coulomb's Law contains
, not simply
.
Doubling the distance makes the force one-quarter as large, not one-half.
Measuring Distance Between the Surfaces
For point-charge calculations involving spherical objects, distance is normally measured centre to centre.
Treating a Negative Direction as a Negative Force Magnitude
Force magnitude is positive.
A negative sign in a coordinate calculation indicates direction.
Ignoring Direction When Adding Forces
Electric force is a vector.
Two forces of 5 N do not necessarily produce a net force of 10 N.
If they act in opposite directions, they can partially or completely cancel.
Confusing Attraction and Repulsion
Remember:
Like charges → repel
Opposite charges → attract
Calculating Only One Force in a Multi-Charge System
Every relevant charge contributes to the net force.
Use the principle of superposition.
Check Your Understanding
1. State Coulomb's Law in words.
2. What determines whether two charged objects attract or repel?
3. What happens to the electric force if one charge is doubled?
4. What happens if both charges are doubled?
5. What happens to the electric force if the distance between two charges doubles?
6. What happens if the distance triples?
7. Convert
into coulombs.
8. Two positive charges are placed near one another. Is the force attractive or repulsive?
9. A positive charge and a negative charge are placed near one another. Is the force attractive or repulsive?
10. Why must electric forces be treated as vectors?
11. Explain the principle of superposition.
12. Two equal positive charges are equal distances on opposite sides of a third positive charge. What is the net force on the centre charge?
13. A charge experiences a force of 8 N right and 3 N left. What is the net force?
14. A charge experiences 6 N right and 8 N upward. Calculate the magnitude of the resultant force.
15. Two charges are moved from a separation of
to
. By what factor does their electric force change?
Key Terms
- Electric charge – a property of matter responsible for electric interactions.
- Coulomb's Law – the relationship describing the electric force between point charges.
- Electric force – an attractive or repulsive force between charged objects.
- Coulomb (C) – the SI unit of electric charge.
- Point charge – an idealized charge treated as being concentrated at a single location.
- Coulomb constant () – the proportionality constant used in Coulomb's Law.
- Attraction – a force that draws objects toward one another.
- Repulsion – a force that pushes objects away from one another.
- Inverse-square law – a relationship in which a quantity varies as.
- Vector – a quantity with both magnitude and direction.
- Net force – the vector sum of all forces acting on an object.
- Superposition – the principle that the net electric force is the vector sum of the individual electric forces.
- Force component – the part of a force acting along a chosen direction.
Key Takeaways
- Like charges repel, while opposite charges attract.
- Coulomb's Law describes how electric force depends on charge magnitude and separation distance.
- Electric force increases as charge magnitude increases.
- Electric force decreases according to an inverse-square relationship as distance increases.
- Doubling one charge doubles the force.
- Doubling both charges makes the force four times larger.
- Doubling the separation makes the force one-quarter as large.
- Charges should be converted into coulombs and distances into metres before using Coulomb's Law.
- Electric force is a vector, so direction must be considered.
- With three or more charges, calculate each individual force and combine them using the principle of superposition.
- Forces along one dimension can be combined using positive and negative directions.
- Forces in two dimensions can be resolved into horizontal and vertical components and combined using vector addition.
This is the format I'll use going forward: full copyable text + inline Moodle-friendly equations + embedded supporting images. The images can enrich the Moodle page when they transfer successfully, but none of the essential learning content will depend on them.
3. Electric field strength
Learning Outcomes
- I can define an electric field as a region where a charge experiences a force.
- I can define electric field strength as force per unit charge.
- I can calculate electric field strength using appropriate equations.
- I can determine the direction of an electric field around a charged object.
- I can explain the relationship between electric force and electric field strength.
- Electric field: \( E = \frac{F}{q} \)
- Units and dimensions of E
Electric Fields
A charged object can exert a force on another charged object even when the two objects are not touching.
We describe the region around a charged object where another charge experiences an electric force as an electric field.
In simple terms:
An electric field is a region where a charged object experiences an electric force.
The charged object that produces the field is called the source charge.
If another charge is placed in that field, it may experience an attractive or repulsive force.
Electric fields allow us to describe how charges influence the space around them.
Electric Fields and Forces
Suppose a positive source charge is placed by itself.
It produces an electric field throughout the space surrounding it.
If another positive charge is placed nearby, it experiences a repulsive electric force.
If a negative charge is placed nearby, it experiences an attractive electric force.
The electric field exists whether or not we actually place another charge there.
This is an important distinction:
- A source charge creates an electric field.
- Another charge placed in that field can experience an electric force.
Electric Field Strength
The strength of an electric field tells us how much force a charge would experience at a particular location.
Electric field strength is defined as:
Electric field strength = force per unit positive charge
The equation is:
where:
- = electric field strength
- = electric force on the charge
- = charge experiencing the force
The SI unit of electric field strength is newtons per coulomb (N/C).
A large value of
represents a strong electric field.
A small value of
represents a weak electric field.
Worked Example: Calculating Electric Field Strength
A charge of
experiences an electric force of
.
Calculate the electric field strength.
Start with:
Substitute:
Therefore:
The electric field strength is 30,000 N/C, or
.
Rearranging the Electric Field Equation
The equation
can be rearranged to calculate electric force.
Multiplying both sides by
gives:
This equation tells us the force experienced by a charge placed in an electric field.
We can also rearrange to find charge:
Therefore, the three useful forms are:
Electric field strength:
Electric force:
Charge:
Worked Example: Calculating Electric Force
A positive charge of
is placed in an electric field with strength
.
Calculate the electric force.
Use:
Substitute:
Therefore:
Because the charge is positive, the force acts in the same direction as the electric field.
Direction of an Electric Field
Electric field direction is defined using a positive test charge.
The direction of an electric field at a point is:
the direction that a positive test charge would be pushed if it were placed at that point.
This convention is extremely important.
Around a positive source charge, a positive test charge would be repelled.
Therefore:
Electric fields point away from positive charges.
Around a negative source charge, a positive test charge would be attracted.
Therefore:
Electric fields point toward negative charges.
Positive Source Charge
Imagine a positive point charge.
Because another positive charge would be repelled, the electric field points radially outward in every direction.
In text form:
← + →
Above the charge, the field points upward.
Below the charge, the field points downward.
To the left, it points left.
To the right, it points right.
The complete three-dimensional field extends outward in all directions.
Negative Source Charge
Now imagine a negative point charge.
A positive test charge would be attracted toward it.
Therefore, the electric field points inward toward the charge.
In text form:
→ − ←
The field direction is always toward the negative source charge.
Electric Field Lines
Electric fields are often represented using electric field lines.
These lines are a visual model of the field.
The arrows on the lines indicate the direction of the electric field.
Remember:
Field lines leave positive charges and enter negative charges.
The spacing of field lines can also indicate field strength.
Where the lines are closer together, the field is stronger.
Where the lines are farther apart, the field is weaker.
Field lines are a model. They do not represent physical strings or tracks in space.
Electric Field Around a Point Charge
The electric field produced by a point charge becomes weaker as distance from the charge increases.
For a point charge, the field strength can be calculated using:
where:
- E = electric field strength in N/C
- k = Coulomb constant
- Q = source charge in C
- r = distance from the source charge in m
The Coulomb constant is approximately:
Notice that this equation uses Q
for the source charge producing the field.
Worked Example: Field Around a Point Charge
A point charge of
produces an electric field.
Calculate the field strength at a distance of
.
Use:
Substitute:
Square the distance:
Then:
Therefore:
The electric field strength is approximately
.
Because the source charge is positive, the field points away from the charge.
How Distance Affects Electric Field Strength
For a point charge:
This means electric field strength follows an inverse-square relationship with distance.
If the distance from the source charge doubles:
So:
Distance × 2 → Field strength × 1/4
If the distance triples:
Distance × 3 → Field strength × 1/9
If the distance is halved:
Distance × 1/2 → Field strength × 4
The electric field therefore becomes much stronger as you move closer to a point charge.
Worked Example: Changing Distance
The electric field strength at a certain distance from a point charge is 800 N/C.
A student moves to a position twice as far from the charge.
Because field strength follows an inverse-square relationship:
So:
The new electric field strength is 200 N/C.
How Source Charge Affects Electric Field Strength
From the equation
, field strength is directly proportional to the magnitude of the source charge.
Therefore:
Source charge × 2 → Field strength × 2
Source charge × 3 → Field strength × 3
Source charge × 1/2 → Field strength × 1/2
A larger source charge produces a stronger electric field at the same distance.
Electric Force and Electric Field Strength
Electric field strength and electric force are closely related, but they are not the same quantity.
Electric field strength describes a property of the field at a particular location.
Electric force describes what happens to a particular charge placed at that location.
The relationship is:
This means that the force depends on:
- the electric field strength
- the size of the charge placed in the field
For the same electric field, a larger charge experiences a larger force.
Worked Example: Same Field, Different Charges
Suppose the electric field strength is:
A charge of
is placed in the field.
The force is:
So:
Now replace it with a charge of
.
The new force is:
The second charge is three times larger, so it experiences three times the force.
However, the electric field strength has not changed.
The field is a property of the location created by the source charges.
Positive and Negative Charges in an Electric Field
The direction of the electric field is defined using a positive test charge.
Therefore, a positive charge experiences force in the same direction as the electric field.
Positive charge:
Electric field →
Force →
A negative charge behaves differently.
Because
is negative, the force on a negative charge acts in the opposite direction to the electric field.
Negative charge:
Electric field →
Force ←
This distinction is extremely important.
Worked Example: Direction of Force
An electric field points to the right with strength
.
A positive charge of
is placed in the field.
The force magnitude is:
The force is 0.020 N to the right because the charge is positive.
Now imagine the charge is
.
The force has the same magnitude, 0.020 N, but acts to the left because the charge is negative.
Electric Fields Between Opposite Charges
When a positive and negative charge are placed near each other, their electric fields combine.
Field lines:
- leave the positive charge
- curve through the surrounding space
- enter the negative charge
This arrangement is called an electric dipole.
The field is not equally strong everywhere.
The strength and direction depend on the location.
Electric Fields Between Like Charges
Two positive charges both produce outward-pointing electric fields.
Between the charges, the two fields point in opposite directions.
If the charges are equal and the point is exactly halfway between them, the fields can cancel.
The net electric field at that point is then:
Two negative charges can produce a similar cancellation point.
This is an example of the principle of superposition: electric fields from different source charges combine as vectors.
Electric Field Strength and Coulomb's Law
There is a direct connection between electric field strength and Coulomb's Law.
Coulomb's Law gives the force between two point charges:
Electric field strength is:
Substituting the Coulomb force into the electric-field equation gives:
The test charge
cancels:
This shows something important:
The electric field produced by a source charge does not depend on the size of the test charge used to measure it.
The field depends on:
- the source charge
- the distancefrom the source charge
Electric Fields Are Vectors
Electric field strength is a vector quantity.
It has:
- magnitude
- direction
If several charged objects produce fields at the same location, the individual fields must be added as vectors.
For fields along the same straight line, choose one direction as positive.
For example:
Field from Charge A = 600 N/C right
Field from Charge B = 250 N/C left
Then:
The net electric field is 350 N/C to the right.
Electric Field vs Electric Force
It is useful to keep these two ideas separate.
Electric field strength:
- Describes the field at a location.
- Measured in N/C.
- Exists even if no test charge is present.
- Depends on the source charges and position.
Electric force:
- Acts on a particular charge.
- Measured in newtons.
- Depends on both the electric field and the charge placed in it.
- Can be calculated using F = qE
- A useful way to remember the difference is:
The field describes the environment.
The force describes what happens to a charge placed in that environment.
Real-World Applications of Electric Fields
Electric fields are important in many technologies and natural phenomena.
Examples include:
- Photocopiers
- Laser printers
- Electrostatic paint spraying
- Electrostatic precipitators
- Capacitors
- Lightning
- Particle accelerators
- Electronic devices
- Charged-particle detectors
Electric fields are also fundamental to understanding interactions between charged particles at atomic and subatomic scales.
Common Mistakes
Confusing Electric Field with Electric Force
They are related but not identical.
Electric field strength is measured in N/C.
Electric force is measured in N.
Forgetting the Meaning of Field Direction
Electric field direction is defined as the direction a positive test charge would experience a force.
Thinking Field Lines Point Toward Positive Charges
Remember:
Away from positive.
Toward negative.
Assuming a Negative Charge Moves in the Field Direction
A negative charge experiences force opposite to the electric field direction.
Forgetting to Convert Charge Units
For example:
Do not substitute 4.0 directly when the equation requires coulombs.
Forgetting to Square Distance
For a point charge:
Doubling distance makes the field one-quarter as strong, not one-half.
Thinking the Test Charge Creates the Field Being Measured
The source charge creates the field. The test charge is used to determine how that field would affect a charge placed there.
Check Your Understanding
1. Define an electric field.
2. Define electric field strength.
3. State the equation relating electric field strength, force, and charge.
4. What is the SI unit of electric field strength?
5. In which direction does the electric field point around a positive point charge?
6. In which direction does the electric field point around a negative point charge?
7. A charge of
experiences a force of 0.040 N. Calculate the electric field strength.
8. A positive charge is placed in an electric field pointing left. In which direction is the force on the charge?
9. A negative charge is placed in the same field. In which direction is its force?
10. What happens to the electric field strength around a point charge if the distance is doubled?
11. What happens to the field strength if the source charge is doubled?
12. Explain the difference between electric field strength and electric force.
13. Why can an electric field exist even when no test charge is present?
14. Two electric fields at a point are 900 N/C right and 400 N/C left. Calculate the net electric field.
15. Explain why a positive and negative charge placed at the same point in an electric field experience forces in opposite directions.
Key Terms
- Electric field – a region in which a charged object experiences an electric force.
- Electric field strength – electric force per unit positive charge at a point.
- Source charge – a charge that produces an electric field.
- Test charge – a small positive charge used conceptually to determine electric field strength and direction.
- Electric force – an attractive or repulsive force acting on a charged object.
- Field line – a line used to represent the direction and relative strength of an electric field.
- Point charge – an idealized charge treated as being concentrated at a single point.
- Newton per coulomb (N/C) – the SI unit of electric field strength.
- Inverse-square relationship – a relationship in which a quantity varies as.
- Electric dipole – a pair of opposite charges separated by a distance.
- Superposition – the principle that individual electric fields combine through vector addition.
- Vector – a quantity with both magnitude and direction.
Key Takeaways
- An electric field is a region where a charge experiences an electric force.
- Electric field strength measures force per unit positive charge.
- Electric field strength can be calculated using.
- Electric force can be calculated using.
- The field around a positive charge points away from the charge.
- The field around a negative charge points toward the charge.
- A positive charge experiences force in the direction of the field.
- A negative charge experiences force opposite to the field direction.
- The electric field around a point charge can be calculated using.
- Electric field strength decreases according to an inverse-square relationship with distance.
- Electric field strength is a vector, so fields produced by multiple charges must be combined using vector addition.
- The electric field describes the electrical environment at a location, while electric force describes how a particular charge responds to that field.
4. Field lines
Learning Outcomes
- I can interpret electric field diagrams using field lines.
- I can describe the direction of electric field lines around positive and negative charges.
- I can explain how field line density indicates field strength.
- I can sketch electric field patterns for isolated and interacting charges.
- I can use field line diagrams to compare electric field strengths in different regions.
Representing an Electric Field
An electric field exists around any charged object. If another charged object is placed in this field, it experiences an electric force.
Electric fields are invisible, so physicists use electric field lines to represent them.
Field lines help us visualize two important properties of an electric field:
- The direction of the field
- The relative strength of the field
The lines themselves are not physical objects. They are a model that helps us understand how a charge would behave at different positions in the field.
Direction of Electric Field Lines
The direction of an electric field is defined as the direction in which a positive test charge would experience a force.
This gives us a simple rule:
Electric field lines point away from positive charges and toward negative charges.
For a positive charge:
Positive charge → field points outward
For a negative charge:
Negative charge → field points inward
This convention applies even if the particle actually placed in the field is negatively charged.
Field Lines Around a Positive Charge
An isolated positive point charge produces a field that extends outward in all directions.
The field has radial symmetry.
This means that the field pattern looks the same in every direction around the charge.
In a sketch:
- Place the positive charge at the centre.
- Draw several lines extending outward.
- Put arrowheads pointing away from the charge.
- Spread the lines evenly around the charge.
As the distance from the charge increases, the electric field becomes weaker.
Field Lines Around a Negative Charge
An isolated negative point charge also produces a field with radial symmetry.
The important difference is the direction.
In a sketch:
- Place the negative charge at the centre.
- Draw several lines approaching the charge from different directions.
- Put arrowheads pointing toward the charge.
- Spread the lines evenly around the charge.
The field becomes stronger as you approach the charge.
Field Line Density and Field Strength
Electric field diagrams can also provide information about field strength.
The general rule is:
Closer field lines → stronger electric field
More widely spaced field lines → weaker electric field
This is sometimes described in terms of field line density.
A region containing many field lines close together represents a stronger electric field than a region where the lines are more widely separated.
The number of drawn lines is a convention of the diagram; what matters when interpreting a given diagram is their relative spacing or density.
Why the Field Weakens with Distance
Around an isolated point charge, the field lines spread farther apart as the distance from the charge increases.
This represents the fact that electric field strength decreases with distance.
For a point charge, the relationship is:
\( E \ \alpha \ \frac{1}{r^2} \)
This means electric field strength follows an inverse-square relationship.
If the distance from the charge doubles:
\( E_{\text{new}} = \frac{E}{4} \)
If the distance triples:
\( E_{\text{new}} = \frac{E}{9} \)
The increasingly wide spacing of field lines provides a visual representation of this decreasing field strength.
Reading a Field Diagram
When you are given an electric field diagram, ask two questions.
Which way does the field point?
Look at the arrowheads on the field lines.
Where is the field strongest?
Look for the region where the field lines are most densely packed.
For example, if the lines are tightly packed near a charge and spread farther apart with distance, the field is stronger near the charge and weaker farther away.
Worked Example: Comparing Two Locations
Imagine two points, A and B, near a positive charge.
Point A is close to the charge.
Point B is much farther away.
At Point A, the field lines are relatively close together.
At Point B, the lines are farther apart.
Therefore:
Electric field at A > Electric field at B
A positive test charge placed at A would experience a greater force than the same test charge placed at B.
This follows from:
\( F = qE \)
For the same test charge
, a larger field strength
produces a larger electric force.
Field Lines Between Opposite Charges
Consider a positive charge placed near a negative charge.
Field lines begin at the positive charge and end at the negative charge.
Therefore:
Positive → Negative
The lines curve because the total electric field at each point is produced by both charges.
This arrangement of equal and opposite charges is called an electric dipole.
Understanding the Dipole Pattern
Near the positive charge, the field points generally away from the positive charge.
Near the negative charge, the field points generally toward the negative charge.
Between the two charges, these effects reinforce each other.
A positive test charge placed between the charges would:
- be repelled by the positive charge
- be attracted toward the negative charge
Both forces act generally from the positive charge toward the negative charge.
The electric field between the charges can therefore be relatively strong.
Field Lines Between Two Positive Charges
Now consider two positive charges.
Both charges produce fields pointing outward.
Between the charges:
- the left positive charge produces a field pointing away from itself
- the right positive charge also produces a field pointing away from itself
The two fields therefore oppose each other in the region between the charges.
For two identical positive charges, there is a point exactly halfway between them where the two electric fields have equal magnitudes and opposite directions.
At that point:
\( E_{\text{net}} = 0 \)
Field Lines Between Two Negative Charges
Two negative charges produce a similar overall pattern, but the field lines point toward the charges.
For two identical negative charges, the fields at the midpoint are equal in magnitude and opposite in direction.
Therefore, at the exact midpoint:
\( E_{\text{net}} = 0 \)
The field is zero at that single point even though strong electric fields exist nearby.
Electric Field Lines Never Cross
An important rule is:
Electric field lines never cross.
Why?
At any particular point in space, the electric field has only one net direction.
If two field lines crossed, the crossing point would suggest that the electric field had two different directions at the same location.
That is impossible.
Therefore, correctly drawn electric field lines cannot cross.
Field Lines Do Not Suddenly Stop in Empty Space
Electric field lines generally:
- begin on positive charges and end on negative charges, or
- extend to or from very large distances when the appropriate opposite charge is not shown
They should not simply stop in empty space without a physical reason.
This rule helps identify incorrectly drawn field diagrams.
Field Lines and Electric Force
The direction of a field line tells us the direction of force on a positive test charge.
Therefore:
Positive charge → force in the same direction as the field
However, a negative charge experiences force in the opposite direction.
Therefore:
Negative charge → force opposite to the field direction
For example, suppose the electric field points to the right.
A positive charge experiences a force to the right.
A negative charge experiences a force to the left.
This follows from:
\( \vec{F} = q\vec{E} \)
The negative value of
reverses the direction of the force.
Worked Example: Reading Force from a Field Diagram
Suppose a field diagram shows field lines pointing from left to right.
A positive charge is placed in the field.
Its force will point:
to the right
Now replace it with a negative charge.
Its force will point:
to the left
The electric field itself has not changed.
Only the direction of the force on the test charge has changed.
Comparing Field Strengths
Suppose a field diagram contains three locations: A, B, and C.
At A, the field lines are tightly packed.
At B, they are moderately spaced.
At C, they are widely spaced.
The relative field strengths are:
A strongest
B intermediate
C weakest
You do not necessarily need numerical values to make this comparison.
The field-line pattern itself provides information about relative field strength.
Uniform Electric Fields
Not all electric fields spread outward from point charges.
A nearly uniform electric field can exist between two large, oppositely charged parallel plates.
In a uniform field:
- field lines are straight
- field lines are parallel
- field lines are approximately equally spaced
- field direction is the same throughout the central region
- field strength is approximately constant throughout that region
The lines point from the positive plate toward the negative plate.
Uniform vs. Non-Uniform Fields
A uniform electric field has approximately the same magnitude and direction throughout a region.
It can be represented by:
- parallel field lines
- equal spacing between lines
A non-uniform electric field changes with position.
The field around a point charge is non-uniform because:
- its direction changes depending on position
- its strength decreases with distance
The field lines therefore spread apart as they move away from the charge.
Sketching Electric Field Patterns
When drawing electric field diagrams, follow these rules:
- Draw arrows away from positive charges.
- Draw arrows toward negative charges.
- Use closer line spacing to represent stronger fields.
- Use wider line spacing to represent weaker fields.
- Never allow field lines to cross.
- For isolated point charges, use a symmetrical radial pattern.
- For interacting charges, allow the lines to curve according to the combined field.
- For uniform fields, draw straight, parallel, equally spaced lines.
The goal is not simply to create a neat picture. The diagram should communicate the direction and relative strength of the electric field.
Sketching an Isolated Positive Charge
Start by drawing:
( + )
Draw lines radiating outward in several directions.
Add arrowheads pointing away from the charge.
Keep the pattern symmetrical.
This represents the field produced by an isolated positive point charge.
Sketching an Isolated Negative Charge
Start with:
( − )
Draw radial lines surrounding the charge.
Place the arrowheads pointing toward the charge.
Again, keep the pattern symmetrical.
Sketching Opposite Charges
Start with:
( + ) ( − )
Draw field lines beginning at the positive charge and ending at the negative charge.
The lines between the charges should point:
+ → −
Additional lines should curve around the outside while still travelling from the positive charge toward the negative charge.
Sketching Two Like Charges
For two positive charges:
( + ) ( + )
Field lines leave both charges.
Between the charges, the fields oppose each other.
The lines bend away from the central region rather than joining one positive charge to the other.
For equal charges, the exact midpoint is a zero-field point.
For two negative charges, the pattern has a similar shape, but the arrows point inward toward the charges.
Principle of Superposition
When several charges are present, each charge produces its own electric field.
The actual electric field at a point is the vector sum of all these fields.
This is called the principle of superposition.
In one dimension:
\( E_{\text{net}} = E_1 + E_2 + E_3 + \cdots \)
Direction must be considered.
For example:
Field from Charge A = 700 N/C right
Field from Charge B = 300 N/C left
Choose right as positive:
\( E_{\text{net}} = +700 - 300 = +400\text{ N/C} \)
Therefore, the net electric field is 400 N/C to the right.
Field Lines as a Model
Field lines are extremely useful, but they are only a model.
There are not actual lines floating through space around a charged object.
A diagram might show 8, 12, or 20 lines around a charge, but the electric field exists throughout the surrounding region, not only along those drawn lines.
Field lines allow us to visualize:
- direction
- relative strength
- symmetry
- interactions between fields
The number of lines drawn is chosen to make the diagram useful and readable.
Common Mistakes
Drawing Field Lines Toward Positive Charges
Remember:
Positive → field lines point away
Negative → field lines point toward
Thinking Field Lines Are Physical Objects
Field lines are a visual model of an electric field.
Allowing Field Lines to Cross
Field lines cannot cross because the net electric field cannot have two different directions at the same point.
Ignoring Line Spacing
Field line density provides information about field strength.
Closer lines = stronger field
Wider spacing = weaker field
Assuming a Negative Charge Moves in the Direction of the Field
The field direction is defined using a positive test charge.
A negative charge experiences a force opposite to the field direction.
Connecting Two Positive Charges with Field Lines
Field lines do not normally begin on one positive charge and end on another positive charge.
They point away from both positive charges.
Assuming Zero Field Means No Fields Are Present
At a cancellation point, several electric fields may be present but their vector sum is zero.
Check Your Understanding
1. What information do electric field lines provide?
2. In which direction do field lines point around a positive charge?
3. In which direction do field lines point around a negative charge?
4. What does closely spaced field-line density indicate?
5. Why do field lines spread farther apart with increasing distance from an isolated point charge?
6. Why can electric field lines never cross?
7. Describe the field pattern around an isolated positive point charge.
8. Describe the field pattern around an isolated negative point charge.
9. In which direction do field lines travel between a positive and negative charge?
10. What happens to the electric fields at the exact midpoint between two identical positive charges?
11. How would you identify the strongest region in a field-line diagram?
12. Describe the field-line pattern in a uniform electric field.
13. A field points right. In which direction would a positive charge experience a force?
14. A field points right. In which direction would a negative charge experience a force?
15. Explain why field lines are considered a model rather than physical objects.
Key Terms
- Electric field – a region in which a charged object experiences an electric force.
- Electric field line – a line used to represent the direction and relative strength of an electric field.
- Field direction – the direction in which a positive test charge would experience a force.
- Field strength – the force per unit positive charge at a particular point.
- Field line density – the relative closeness of field lines used to represent field strength.
- Radial field – a field whose lines extend directly toward or away from a central point.
- Uniform electric field – a field with approximately constant magnitude and direction throughout a region.
- Non-uniform electric field – a field whose magnitude or direction changes with position.
- Electric dipole – a pair of opposite charges separated by a distance.
- Superposition – the principle that electric fields from different charges combine through vector addition.
- Test charge – a small positive charge used conceptually to determine field direction.
Key Takeaways
- Electric field lines provide a visual representation of electric field direction and relative strength.
- Field lines point away from positive charges.
- Field lines point toward negative charges.
- The electric field is stronger where field lines are more densely packed.
- The electric field is weaker where field lines are more widely spaced.
- Field lines around isolated point charges form radial patterns.
- Field lines between opposite charges travel from the positive charge toward the negative charge.
- Field lines around like charges bend away from the region between the charges.
- Electric field lines never cross.
- A uniform electric field is represented by parallel, equally spaced field lines.
- Field diagrams can be used to compare electric field strengths without knowing numerical values.
- Field lines are a model; they are not physical structures in space.
5. Superposition of fields
Learning Outcomes
- I can explain the principle of superposition for electric fields.
- I can determine the net electric field produced by multiple charges.
- I can add electric field vectors using vector addition techniques.
- I can identify locations where electric fields reinforce or cancel each other.
- I can solve problems involving electric fields created by multiple charges.
When More Than One Charge Creates a Field
A single charged object produces an electric field around itself. In many real situations, however, there are several charges present at the same time.
Each charge produces its own electric field.
At any particular point, these individual electric fields combine to produce a net electric field.
The rule used to combine them is called the principle of superposition.
The Principle of Superposition
The principle of superposition states:
The net electric field at a point is the vector sum of the electric fields produced by all individual source charges.
In copyable Moodle TeX:
\( \vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \cdots \)
This means we can calculate the field produced by each charge separately, then combine those fields using vector addition.
A useful procedure is:
- Choose the point where you want to determine the electric field.
- Calculate the field produced by each source charge at that point.
- Determine the direction of each field.
- Add the electric field vectors.
- State both the magnitude and direction of the net field.
Electric Field Produced by a Point Charge
The magnitude of the electric field produced by a point charge can be calculated using:
\( E = k\frac{|Q|}{r^2} \)
where:
- = electric field strength in N/C
- = Coulomb constant
- = source charge in C
- = distance from the source charge in m
Use:
\( k = 8.99\times10^9\text{ N m}^2/\text{C}^2 \)
The equation gives the magnitude of the field.
The sign of the source charge determines the direction.
Remember:
Positive source charge → field points away from the charge
Negative source charge → field points toward the charge
Why Direction Matters
Electric field is a vector quantity.
This means that simply adding the numerical field strengths is not always correct.
Suppose two charges produce fields at Point P:
Field 1 = 500 N/C right
Field 2 = 300 N/C left
The fields oppose each other.
Therefore:
\( E_{net} = 500 - 300 = 200\text{ N/C} \)
The net field is 200 N/C to the right.
If both fields pointed right instead, we would add them:
\( E_{net} = 500 + 300 = 800\text{ N/C} \)
So determining direction should normally come before adding the field magnitudes.
Fields That Reinforce Each Other
Electric fields reinforce when their vectors point in the same direction.
For example:
Field 1 → 400 N/C
Field 2 → 600 N/C
Both point right.
Therefore:
\( E_{net} = 400 + 600 = 1000\text{ N/C} \)
The net field is 1000 N/C to the right.
Reinforcement produces a stronger net field.
Fields That Oppose Each Other
Electric fields oppose each other when their vectors point in opposite directions.
For example:
Field 1 → 900 N/C
Field 2 ← 400 N/C
Choose right as positive:
\( E_{net} = +900 - 400 = +500\text{ N/C} \)
Therefore, the net field is 500 N/C to the right.
The larger field determines the final direction.
Complete Cancellation
If two fields have equal magnitudes and opposite directions, they cancel completely.
For example:
Field 1 → 600 N/C
Field 2 ← 600 N/C
Then:
\( E_{net} = 600 - 600 = 0\text{ N/C} \)
This is called a zero-field point or null point.
It is important to understand that the individual fields have not disappeared.
Both fields exist, but their vector sum is zero.
Two Equal Positive Charges
Consider two identical positive charges:
( +Q ) -------- P -------- ( +Q )
Suppose Point P is exactly halfway between them.
The left positive charge produces a field at P pointing to the right because electric fields point away from positive charges.
The right positive charge produces a field at P pointing to the left.
Because:
- the charges are equal
- the distances are equal
the field magnitudes are equal.
Therefore:
\( E_1 = E_2 \)
but their directions are opposite.
So:
\( E_{net} = 0 \)
The midpoint is a zero-field point.
Two Equal Negative Charges
The same cancellation can occur between two identical negative charges.
Consider:
( −Q ) -------- P -------- ( −Q )
The left negative charge produces a field pointing left because the field points toward the negative charge.
The right negative charge produces a field pointing right.
At the exact midpoint:
\( E_1 = E_2 \)
Therefore:
\( E_{net} = 0 \)
Again, the fields cancel because they have equal magnitudes and opposite directions.
Between Opposite Charges
Now consider equal positive and negative charges:
( +Q ) -------- P -------- ( −Q )
At Point P:
- The positive charge produces a field pointing away from itself → right.
- The negative charge produces a field pointing toward itself → right.
Both fields therefore point in the same direction.
So they reinforce:
\( E_{net} = E_1 + E_2 \)
This is why the field between opposite charges can be particularly strong.
Worked Example: Two Equal Positive Charges
Two positive charges each have a charge of:
\( +4.0\times10^{-6}\text{ C} \)
They are separated by 0.40 m.
Find the net electric field at the midpoint.
The midpoint is 0.20 m from each charge.
Calculate the field produced by one charge:
\( E = k\frac{|Q|}{r^2} \)
Substitute:
\( E = (8.99\times10^9)\frac{4.0\times10^{-6}}{(0.20)^2} \)
Since:
\( (0.20)^2 = 0.040 \)
we obtain:
\( E = 8.99\times10^5\text{ N/C} \)
Each charge produces the same field magnitude.
However, the fields point in opposite directions.
Therefore:
\( E_{net} = 8.99\times10^5 - 8.99\times10^5 \)
So:
\( E_{net} = 0\text{ N/C} \)
The electric fields cancel at the midpoint.
Worked Example: Opposite Charges
Suppose the previous right-hand charge is changed to:
\( -4.0\times10^{-6}\text{ C} \)
The two charges are still 0.40 m apart.
At the midpoint, each charge still produces a field of:
\( 8.99\times10^5\text{ N/C} \)
However, the directions have changed.
The positive charge produces a field toward the right.
The negative charge also produces a field toward the right.
Therefore:
\( E_{net} = 8.99\times10^5 + 8.99\times10^5 \)
So:
\( E_{net} = 1.798\times10^6\text{ N/C} \)
Rounded appropriately:
\( E_{net} \approx 1.80\times10^6\text{ N/C} \)
The net field points from the positive charge toward the negative charge.
Unequal Charges
If two charges have different magnitudes, their electric fields will not necessarily cancel at the midpoint.
Suppose:
( +Q ) -------- P -------- ( +2Q )
If P is exactly halfway between them, both fields act over the same distance.
However, the
charge produces twice the field strength because:
\( E \propto Q \)
The field produced by
therefore dominates.
The net field points in the direction of the stronger field at P.
Finding a Zero-Field Point for Unequal Like Charges
For two unequal charges of the same sign, the zero-field point lies somewhere between the charges.
It will not normally be at the midpoint.
Suppose:
( +Q₁ ) ----- P ---------- ( +Q₂ )
At P, the fields point in opposite directions.
For complete cancellation:
\( E_1 = E_2 \)
Therefore:
\( k\frac{|Q_1|}{r_1^2} = k\frac{|Q_2|}{r_2^2} \)
The Coulomb constants cancel:
\( \frac{|Q_1|}{r_1^2} = \frac{|Q_2|}{r_2^2} \)
The zero-field point will be closer to the smaller-magnitude charge.
Why?
Because you must move closer to the weaker source charge to make its field strong enough to equal the field produced by the larger charge.
Worked Example: Locating a Zero-Field Point
Suppose two positive charges are:
\( Q_1 = +1.0\ \mu\text{C} \)
and
\( Q_2 = +4.0\ \mu\text{C} \)
They are 3.0 m apart.
Let the zero-field point be a distance
from the smaller charge.
Its distance from the larger charge is:
\( 3.0-x \)
For cancellation:
\( \frac{1.0}{x^2} = \frac{4.0}{(3.0-x)^2} \)
Taking the positive square root:
\( \frac{1}{x} = \frac{2}{3.0-x} \)
Cross-multiply:
\( 3.0-x = 2x \)
Therefore:
\( 3.0 = 3x \)
So:
\( x = 1.0\text{ m} \)
The zero-field point is 1.0 m from the smaller charge and 2.0 m from the larger charge.
Superposition in Two Dimensions
Electric fields are not always arranged along a straight line.
Suppose two fields act at Point P:
Field 1 = 300 N/C right
Field 2 = 400 N/C upward
Because the fields are perpendicular, we cannot simply add 300 and 400.
Instead, use vector addition.
The magnitude of the resultant field is:
\( E_{net} = \sqrt{E_x^2 + E_y^2} \)
Substitute:
\( E_{net} = \sqrt{(300)^2 + (400)^2} \)
So:
\( E_{net} = \sqrt{90000 + 160000} \)
Therefore:
\( E_{net} = \sqrt{250000} = 500\text{ N/C} \)
The net electric field strength is 500 N/C.
Finding the Direction of the Net Field
For the previous example:
Horizontal field = 300 N/C right
Vertical field = 400 N/C upward
The direction can be calculated using:
\( \tan\theta = \frac{E_y}{E_x} \)
Substitute:
\( \tan\theta = \frac{400}{300} \)
Therefore:
\( \theta = \tan^{-1}\left(\frac{400}{300}\right) \)
So:
\( \theta \approx 53^\circ \)
The net electric field is therefore:
500 N/C at approximately 53° above the horizontal.
Resolving Electric Fields into Components
When fields act at angles, it is often easier to resolve each field into horizontal and vertical components.
For a field
making an angle
with the horizontal:
\( E_x = E\cos\theta \)
and:
\( E_y = E\sin\theta \)
After resolving every field, add the horizontal components:
\( E_{x,net} = E_{1x} + E_{2x} + E_{3x} + \cdots \)
Then add the vertical components:
\( E_{y,net} = E_{1y} + E_{2y} + E_{3y} + \cdots \)
Finally:
\( E_{net} = \sqrt{E_{x,net}^2 + E_{y,net}^2} \)
This method works for systems containing many charges.
Worked Example: Adding Components
Suppose three charges create the following fields at Point P:
- 500 N/C right
- 200 N/C left
- 400 N/C upward
Choose right and upward as positive.
Horizontal component:
\( E_{x,net} = +500 - 200 = +300\text{ N/C} \)
Vertical component:
\( E_{y,net} = +400\text{ N/C} \)
Now calculate the resultant:
\( E_{net} = \sqrt{300^2 + 400^2} \)
Therefore:
\( E_{net} = 500\text{ N/C} \)
The direction is:
\( \theta = \tan^{-1}\left(\frac{400}{300}\right) \approx 53^\circ \)
So the net field is 500 N/C at 53° above the horizontal toward the right.
Three or More Source Charges
The principle of superposition works for any number of source charges.
Suppose four charges produce electric fields at Point P.
Then:
\( \vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \vec{E}_4 \)
You do not need a new law for four charges.
Instead:
- Calculate.
- Calculate.
- Calculate.
- Calculate.
- Determine each direction.
- Resolve into components if necessary.
- Add all components.
- Calculate the resultant.
This is one of the great advantages of the principle of superposition.
A Reliable Problem-Solving Method
For multi-charge electric field problems, use the following sequence.
Choose the Point of Interest
Identify the exact point where the net field is required.
Draw the Charge Arrangement
Mark:
- Each charge
- Its sign
- Relevant distances
- The point where the field is being calculated
Determine Directions Before Calculating
For each source charge ask:
If I placed a positive test charge at the point, which way would it be pushed?
Remember:
Away from positive
Toward negative
Calculate Each Field Separately
Use:
\( E = k\frac{|Q|}{r^2} \)
Add the Fields as Vectors
For one-dimensional problems, use positive and negative directions.
For two-dimensional problems, use components.
State Magnitude and Direction
A complete vector answer should normally include both.
Using Field-Line Diagrams to Recognize Superposition
Field-line diagrams provide a visual representation of superposition.
With opposite charges, field lines often connect the positive and negative charges because the fields between them reinforce.
With like charges, field lines bend away from the region between the charges because the fields oppose each other there.
For equal like charges, a cancellation point occurs at the midpoint.
Superposition and Electric Force
Superposition can be applied to both electric fields and electric forces.
Once the net electric field is known, the force on a charge placed there can be found using:
\( \vec{F} = q\vec{E}_{net} \)
For a positive charge, the force points in the same direction as the net electric field.
For a negative charge, the force points in the opposite direction.
For example, if:
\( E_{net} = 2000\text{ N/C right} \)
and a positive charge of:
\( q = 3.0\times10^{-6}\text{ C} \)
is placed there, then:
\( F = qE \)
\( F = (3.0\times10^{-6})(2000) \)
So:
\( F = 6.0\times10^{-3}\text{ N} \)
The force is 0.0060 N to the right.
Common Mistakes
Adding Magnitudes Without Considering Direction
Electric fields are vectors.
500 N/C right + 300 N/C left is not 800 N/C.
The correct result is 200 N/C right.
Using the Sign of the Source Charge Inside the Magnitude Calculation
Use the magnitude of the charge to calculate field strength.
Then determine direction separately using:
Away from positive
Toward negative
Assuming Equal Charges Always Cancel
Equal charges only produce cancelling fields at locations where their field vectors are equal and opposite.
Assuming the Midpoint Is Always a Zero-Field Point
The midpoint is a zero-field point between equal like charges, but not between equal opposite charges.
Forgetting the Inverse-Square Relationship
Field strength depends on:
\( \frac{1}{r^2} \)
A charge twice as far away does not produce half the field. It produces one-quarter of the field.
Adding Perpendicular Fields Arithmetically
If one field is 300 N/C right and another is 400 N/C upward, the result is not 700 N/C.
Vector addition gives 500 N/C.
Forgetting Direction in the Final Answer
Electric field is a vector. A complete answer should include direction whenever the problem requires it.
Check Your Understanding
1. State the principle of superposition for electric fields.
2. Two fields at a point are 700 N/C right and 200 N/C right. What is the net field?
3. Two fields are 700 N/C right and 200 N/C left. What is the net field?
4. Two fields are 450 N/C in opposite directions. What is the net field?
5. Why is the electric field zero at the midpoint between two identical positive charges?
6. Why do the electric fields reinforce between equal positive and negative charges?
7. Two perpendicular electric fields have strengths of 600 N/C and 800 N/C. Calculate the magnitude of the net field.
8. A positive charge is placed at a point where the net electric field points left. In which direction will the electric force act?
9. How would your answer to Question 8 change if the charge were negative?
10. Why must electric fields be added as vectors rather than ordinary numbers?
11. Where is the zero-field point located between two unequal positive charges: closer to the larger charge or closer to the smaller charge? Explain.
12. A field of 900 N/C right combines with a field of 500 N/C left and a field of 300 N/C right. Determine the net field.
13. Explain why two equal opposite charges do not produce a zero electric field at their midpoint.
14. Describe a systematic method for solving a problem involving three source charges.
15. Explain how a field-line diagram can show regions where electric fields reinforce or oppose each other.
Key Terms
- Superposition – the principle that the net electric field is the vector sum of the individual electric fields.
- Net electric field – the combined electric field produced by all source charges at a point.
- Source charge – a charge that produces an electric field.
- Vector addition – the process of combining quantities that have both magnitude and direction.
- Resultant – the single vector representing the combined effect of several vectors.
- Component – the part of a vector acting along a chosen direction.
- Reinforcement – a situation in which electric field vectors act in the same direction and increase the net field.
- Cancellation – a situation in which opposing electric field vectors reduce or eliminate the net field.
- Zero-field point – a location where the vector sum of the electric fields is zero.
- Electric field strength – electric force per unit positive charge at a point.
Key Takeaways
- Every source charge produces its own electric field.
- The principle of superposition allows these individual fields to be combined.
- Electric fields are vectors, so both magnitude and direction matter.
- Fields pointing in the same direction reinforce each other.
- Fields pointing in opposite directions partially or completely cancel.
- Equal like charges produce a zero-field point at their midpoint.
- Equal opposite charges produce fields that reinforce between the charges rather than cancel.
- For unequal like charges, the zero-field point lies closer to the smaller-magnitude charge.
- One-dimensional fields can be added using positive and negative directions.
- Two-dimensional fields can be resolved into horizontal and vertical components.
- For perpendicular components, the resultant magnitude can be found using Pythagoras.
- The net electric field can be used to calculate the force on a charge using
\( \vec{F} = q\vec{E}_{net} \). - Superposition works for systems containing two, three, or many charges.