Coulomb's Law

Electric charges exert forces on one another even when they are not touching. These forces are known as electrostatic forces and can be either attractive or repulsive depending on the types of charges involved. Positive and negative charges attract each other, while two positive charges or two negative charges repel each other.

The strength of the electric force between two point charges is described by Coulomb's Law, which was developed by the French physicist Charles-Augustin de Coulomb in the late eighteenth century. Coulomb's Law states that the electric force between two charges is:

• directly proportional to the product of the charges,
• inversely proportional to the square of the distance between them.

The mathematical form of Coulomb's Law is:

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

​

where:

• F = electric force (N)
• k = 8.99•10⁹ Nm²/C² (Coulomb's constant)
• k=8.99×109N\cdotpm2/C2 
• q₁​ and q₂​ = charges (C)
• r = separation distance between the charges (m)

The absolute value signs indicate that the equation calculates the magnitude of the force. The direction of the force must be determined separately by considering whether the charges attract or repel.

Attractive Force

Two charges of +2.0•10⁻⁶ C and −3.0•10⁻⁶ C are separated by 0.50m.

Substituting into Coulomb's Law:

\( F = \frac{(8.99×109)|(2.0×10^{−6})(3.0×10^{−6})|}{(0.5)^2} = 0.216 N \)

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

Repulsive Force

Two electrons are separated by a distance of 1.0×10⁻⁹ m.

Because both charges are negative, the force between them is repulsive. Coulomb's Law can be used to calculate the magnitude of this force.

One of the most important features of Coulomb's Law is the inverse square relationship. Because distance appears squared in the denominator, changes in separation distance have a large effect on the force.

Effect of Distance

If the distance between two charges is doubled: r → 2r

then:

\( F \rightarrow \frac{F}{4} \)

​

The force becomes one-quarter as large.

If the distance is tripled: r→3r

then:

\( F \rightarrow \frac{F}{9} \)

​

The force becomes one-ninth as large.

The force also depends directly on the size of the charges.

Effect of Charge

If one charge is doubled while the distance remains constant: q_1​ → 2q₁​ then:

F → 2F

The force doubles.

If both charges are doubled:  q_1​ → 2q₁​ and q₂​ → 2q₂​ then:

F → 4F

The force becomes four times larger.

In many situations, a charge experiences forces from several other charges simultaneously. To determine the total force, we use the principle of superposition. This principle states that the net electric force is the vector sum of all individual electric forces.

\( F_{net} = F_1 + F_2 + F_2 + ... \)

Three Charges on a Line

Suppose a positive charge is located between two other positive charges.

1. Calculate the force exerted by the charge on the left.
2. Calculate the force exerted by the charge on the right.
3. Determine the direction of each force.
4. Add the forces using vector addition.

The resulting force is the net electric force acting on the charge.

For systems arranged in two dimensions, forces must be resolved into horizontal and vertical components before being combined. This often requires trigonometry and vector addition techniques.

Charges at the Corners of a Triangle

Three charges are placed at the corners of a right triangle. The force on one charge is determined by:

• calculating the force from each of the other two charges,
• resolving forces into components,
• adding the horizontal and vertical components,
• finding the magnitude and direction of the resultant force.

This approach allows Coulomb's Law to be applied to complex charge configurations and prepares students for the study of electric fields.

Coulomb's Law is one of the fundamental laws of electromagnetism. It explains how charged objects interact and provides the foundation for understanding electric fields, electric potential, capacitors, circuits, and many modern technologies.

Suggested Diagrams

1. Two positive charges repelling each other.
2. A positive and negative charge attracting each other.
3. An inverse-square graph of Force vs Distance.
4. Three charges arranged on a line showing force vectors.
5. Three charges arranged in a triangle showing vector addition of forces.

One of the most important ideas in electricity is that charged objects create an electric field around them. An electric field is an invisible region of space where other charges experience an electric force. Rather than thinking of charged objects as somehow "reaching out" to pull or push one another, physicists describe these interactions through electric fields. Any charged object placed within an electric field will experience a force.

Imagine placing a small positive test charge near a positively charged sphere. The test charge would be repelled and move away from the sphere. This happens because the sphere creates an electric field that exerts a force on the test charge. The stronger the field, the greater the force experienced by the charge.

To measure the strength of an electric field, physicists use a quantity called electric field strength, represented by the symbol E.

Electric field strength is defined as the force experienced per unit charge:

\( E = \frac{F}{q} \)

where:

• E = electric field strength (N/C)
• F = electric force (N)
• q = charge experiencing the force (C)

This equation tells us how much force each coulomb of charge would experience at a particular location in the field.

For example, if a charge of 2.0 C experiences a force of 10 N, the electric field strength at that location is:

\( E = \frac{10}{2.0} = 5.0 N/C \)

This means every coulomb of positive charge placed at that location would experience a force of 5.0 N.

Electric field strength is a vector quantity, meaning it has both magnitude and direction. The direction of the electric field is defined as the direction that a positive test charge would move if placed in the field.

This leads to two important rules:

• Electric fields point away from positive charges.
• Electric fields point toward negative charges.

Because electric fields are vectors, both their strength and direction must be considered when solving problems.

Electric Field Around a Point Charge

The electric field produced by a single point charge can also be calculated directly from the source charge:

\( E = \frac{kQ}{r^2} \)​

where:

• E = electric field strength (N/C)
• k = Coulomb's constant (8.99 × 10⁹ N·m²/C²)
• Q = source charge (C)
• r = distance from the source charge (m)

This equation shows two important relationships:

Larger Charges Produce Stronger Fields

As the charge Q increases, the electric field strength increases.

A charge of +10 μC produces a stronger field than a charge of +1 μC at the same distance.

Electric Fields Weaken with Distance

Electric field strength decreases according to an inverse-square relationship.

If the distance from a charge doubles:

\( E \alpha \frac{1}{r^2} \)​

the electric field becomes four times weaker.

If the distance triples, the field becomes nine times weaker.

This inverse-square behavior is similar to the behavior of gravitational fields.

Relationship Between Force and Electric Field

The electric force experienced by a charge depends on both the electric field strength and the amount of charge present.

Rearranging the electric field equation gives:

F = qE

This equation shows that:

• Larger charges experience larger forces.
• Stronger electric fields produce larger forces.
• The force direction depends on the sign of the charge.

A positive charge experiences force in the direction of the field.

A negative charge experiences force opposite the direction of the field.

Comparing Electric and Gravitational Fields

Electric fields and gravitational fields share many similarities:

While gravity is always attractive, electric fields can either attract or repel depending on the charges involved.

Applications of Electric Fields

Electric fields are used in many modern technologies:

• Photocopiers and laser printers
• Electrostatic precipitators that remove pollution particles
• Particle accelerators
• Cathode-ray tubes
• Electron microscopes

In each case, electric fields are used to exert forces on charged particles and control their motion.

Key Equations

Electric field strength:

\( E = \frac{F}{q} \)

Electric field around a point charge:

\( E = \frac{kQ}{r^2} \)​

In many real-world situations, electric fields are not produced by a single charge. Instead, multiple charged objects may be present, each creating its own electric field. To determine the overall electric field at a particular location, physicists use a principle known as the principle of superposition.

The principle of superposition states that:

The total electric field at a point is equal to the vector sum of all individual electric fields acting at that point.

This means that each charge contributes its own electric field independently, and the combined effect is found by adding the electric field vectors together.

Why Superposition Is Needed

Consider two charges placed near each other.

Each charge creates an electric field that extends throughout the surrounding space.

At any location:

• Charge 1 produces an electric field.
• Charge 2 produces an electric field.
• The total field is the combination of both fields.

Rather than one field replacing the other, both fields exist simultaneously and overlap.

This overlapping of fields is known as superposition.

Electric Fields Are Vectors

Because electric fields have both magnitude and direction, they must be added using vector addition.

This means:

• Fields pointing in the same direction reinforce one another.
• Fields pointing in opposite directions partially or completely cancel.
• Fields at angles must be added using vector methods.

The direction of each field is just as important as its strength.

Fields in the Same Direction

Suppose two positive charges create electric fields at a point that both point to the right.

If:

E₁  = 40 N/C

and

E₂​ = 60 N/C

then:

E_net​ = E₁ ​+ E₂​ = 100 N/C

Because the fields point in the same direction, their magnitudes simply add together.

Fields in Opposite Directions

Now suppose two charges create fields that point in opposite directions.

If:

E1​ = 80 N/C

to the right and

E₂​ = 50 N/C

to the left, then:

E_net​ = 80 − 50 ​= 30 N/C

to the right.

The stronger field determines the final direction.

Equal and Opposite Fields

Sometimes two fields have equal magnitude but opposite direction.

For example:

E₁​ = 75 N/C

to the right and

E₂​ = 75 N/C

to the left.

In this case:

E_net ​= 0

The fields cancel completely.

Such locations are called electric field null points or equilibrium points.

At these locations a positive test charge experiences no net electric force.

Superposition Between Two Like Charges

Consider two identical positive charges.

At the midpoint between them:

• One field points away from the left charge.
• One field points away from the right charge.
• The fields point in opposite directions.

Because the charges are equal, the fields cancel exactly at the midpoint.

The net electric field is zero.

This creates a field null point between the charges.

Superposition Between Opposite Charges

Now consider a positive charge and a negative charge.

At points between the charges:

• The field from the positive charge points away from the positive charge.
• The field from the negative charge points toward the negative charge.

Both fields point in the same direction.

As a result, the fields reinforce one another and the net field becomes stronger.

This helps explain why opposite charges attract so strongly.

Two-Dimensional Superposition

In many situations, electric fields are not perfectly aligned.

For example:

• One field may point east.
• Another may point north.

Because electric fields are vectors, they must be added using vector methods.

Students often use:

• Scale vector diagrams
• Pythagorean theorem
• Trigonometry

to determine the magnitude and direction of the resultant field.

This is very similar to adding force vectors or velocity vectors.

Superposition and Coulomb's Law

When calculating fields from multiple charges, physicists often follow these steps:

Step 1

Calculate the electric field from each charge individually.

Use:

\( E = \frac{kQ}{r^2} \)​

Step 2

Determine the direction of each field.

Remember:

• Away from positive charges
• Toward negative charges

Step 3

Add the field vectors using vector addition.

The result is the net electric field.

Why Superposition Matters

The principle of superposition is one of the most powerful ideas in physics.

It allows scientists and engineers to analyze:

• Electric circuits
• Particle accelerators
• Capacitors
• Electronic devices
• Electric field mapping

Without superposition, calculating fields from multiple charges would be nearly impossible.

The same principle is also used in:

• Gravitational fields
• Magnetic fields
• Waves
• Sound
• Light

Key Equations

Electric field from a point charge:

\( E = \frac{kQ}{r^2} \)

Superposition principle:

E_net = E₁ + E₂ + E₃ + ...

Superposition Outcomes

Suggested Diagrams

Diagram 1: Two electric field vectors pointing in the same direction and adding together.

Diagram 2: Two electric field vectors pointing in opposite directions and partially cancelling.

Diagram 3: Two identical positive charges showing a field null point at the midpoint.

Diagram 4: A positive-negative charge pair showing reinforcement of fields between the charges.

Diagram 5: Two electric field vectors at right angles being added using vector addition.

Summary

The principle of superposition states that the total electric field at a point is the vector sum of all individual electric fields present. Because electric fields are vectors, both magnitude and direction must be considered when combining them. Fields may reinforce, partially cancel, or completely cancel depending on their directions. Superposition allows physicists to analyze complex systems involving multiple charges and is a fundamental principle used throughout electricity, magnetism, waves, and many other areas of physics.