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.