Basics of Electric Fields

2. Coulomb’s Law

 

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.

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Coulomb's Law

For two point charges, the magnitude of the electric force is:

$F = k\frac{|q_1q_2|}{r^2}$

where:

• $F$= magnitude of electric force in newtons (N)
• $q_1$= first charge in coulombs (C)
• $q_2$= second charge in coulombs (C)
• $r$= distance between the centres of the charges in metres (m)
• $k$= Coulomb constant

The Coulomb constant is approximately

$k = 8.99\times10^9\text{ N m}^2/\text{C}^2$

.

The absolute-value signs around

$q_1q_2$

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.

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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

$r$

is normally measured from centre to centre, not from surface to surface.

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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 ($\mu\text{C}$)
• Nanocoulombs ($\text{nC}$)

Remember that

$1\ \mu\text{C} = 1\times10^{-6}\text{ C}$

.

Also,

$1\text{ nC} = 1\times10^{-9}\text{ C}$

.

Charges should normally be converted to coulombs before being substituted into Coulomb's Law.

Worked Example: Converting Charge

Convert

$5.0\ \mu\text{C}$

into coulombs.

Since

$1\ \mu\text{C} = 10^{-6}\text{ C}$

:

$5.0\ \mu\text{C} = 5.0\times10^{-6}\text{ C}$

Similarly:

$250\text{ nC} = 250\times10^{-9}\text{ C} = 2.50\times10^{-7}\text{ C}$

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Calculating Electric Force

Suppose two point charges have values

$q_1 = +2.0\times10^{-6}\text{ C}$

and

$q_2 = -3.0\times10^{-6}\text{ C}$

.

They are separated by

$r = 0.50\text{ m}$

.

Start with Coulomb's Law:

$F = k\frac{|q_1q_2|}{r^2}$

Substitute the values:

$F = (8.99\times10^9)\frac{|(2.0\times10^{-6})(-3.0\times10^{-6})|}{(0.50)^2}$

First multiply the charges:

$|(2.0\times10^{-6})(-3.0\times10^{-6})| = 6.0\times10^{-12}$

Square the distance:

$(0.50)^2 = 0.25$

Therefore:

$F = (8.99\times10^9)\frac{6.0\times10^{-12}}{0.25}$

This gives:

$F \approx 0.216\text{ N}$

Rounded appropriately, the electric force is approximately 0.22 N.

Because the charges have opposite signs, the force is attractive.

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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.

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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:

$F_{\text{A on B}} = F_{\text{B on A}}$

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.

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How Charge Magnitude Affects Force

Coulomb's Law shows:

$F \propto |q_1q_2|$

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:

$F_{\text{new}} = 2(6.0\text{ N}) = 12\text{ N}$

If both charges are doubled:

$F_{\text{new}} = 4(6.0\text{ N}) = 24\text{ N}$

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How Distance Affects Electric Force

Distance has an especially strong effect because it is squared in Coulomb's Law.

$F \propto \frac{1}{r^2}$

This is called an inverse-square relationship.

If the distance doubles:

$F_{\text{new}} = \frac{F}{2^2} = \frac{F}{4}$

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.

$F_{\text{new}} = \frac{12\text{ N}}{2^2} = 3.0\text{ N}$

If their separation were instead reduced to half:

$F_{\text{new}} = 4(12\text{ N}) = 48\text{ N}$

---

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:

$2\times2 = 4$

Doubling the distance changes the force by:

$\frac{1}{2^2} = \frac{1}{4}$

Combining the effects:

$4\times\frac{1}{4} = 1$

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

$1/4$

.

Therefore:

$F_{\text{new}} = 8.0\times3\times\frac{1}{4} = 6.0\text{ N}$

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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.

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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:

$\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots$

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.

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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.

$F_{\text{net}} = +5.0\text{ N} - 3.0\text{ N} = +2.0\text{ N}$

Therefore, the net force is 2.0 N to the right.

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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.

$F_{\text{net}} = 4.0\text{ N} + 6.0\text{ N} = 10.0\text{ N}$

The net force is 10.0 N to the right.

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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:

$F_{\text{net}} = 4.0\text{ N} - 4.0\text{ N} = 0\text{ N}$

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.

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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:

$F_{\text{net}} = \sqrt{F_x^2 + F_y^2}$

Substitute:

$F_{\text{net}} = \sqrt{(3.0)^2 + (4.0)^2}$

So:

$F_{\text{net}} = \sqrt{25} = 5.0\text{ N}$

The resultant force has a magnitude of 5.0 N.

To find its direction:

$\tan\theta = \frac{F_y}{F_x} = \frac{4.0}{3.0}$

Therefore:

$\theta = \tan^{-1}\left(\frac{4.0}{3.0}\right) \approx 53^\circ$

The net force is approximately 5.0 N at 53° above the horizontal.

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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.

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Superposition in Two Dimensions

Add all horizontal components:

$F_{x,\text{net}} = F_{1x} + F_{2x} + F_{3x} + \cdots$

Add all vertical components:

$F_{y,\text{net}} = F_{1y} + F_{2y} + F_{3y} + \cdots$

Then calculate the resultant magnitude:

$F_{\text{net}} = \sqrt{F_{x,\text{net}}^2 + F_{y,\text{net}}^2}$

The direction can be calculated using:

$\theta = \tan^{-1}\left(\frac{F_{y,\text{net}}}{F_{x,\text{net}}}\right)$

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:

$F_e \propto \frac{1}{r^2}$

Gravitational force:

$F_g \propto \frac{1}{r^2}$

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.

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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.

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Common Mistakes

Forgetting to Convert Microcoulombs

If

$q = 4.0\ \mu\text{C}$

, do not substitute 4.0 directly into Coulomb's Law.

Convert first:

$4.0\ \mu\text{C} = 4.0\times10^{-6}\text{ C}$

Forgetting to Square the Distance

Coulomb's Law contains

$r^2$

, not simply

$r$

.

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.

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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

$6.5\ \mu\text{C}$

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

$r$

to

$2r$

. By what factor does their electric force change?

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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 ($k$) – 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$1/r^2$.
• 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.

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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.

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