Electric Charge and Coulomb's Law

Welcome to the foundational concepts of Electrostatics! In this section, we will explore the fundamental property of matter known as electric charge and the law that governs the interaction between these charges: Coulomb's Law. Understanding these principles is crucial for grasping many other topics in physics, from electric fields to electric potential and beyond.

1. Electric Charge

Electric charge is an intrinsic property of matter that causes it to experience a force when placed in an electromagnetic field. It is a scalar quantity, meaning it has magnitude but no direction. There are two types of electric charges:

1.1 Types of Electric Charge

• Positive Charge (+): Associated with protons. When a substance loses electrons, it becomes positively charged. For instance, a glass rod rubbed with silk becomes positively charged.
• Negative Charge (-): Associated with electrons. When a substance gains electrons, it becomes negatively charged. For example, an ebonite rod rubbed with fur becomes negatively charged.

The fundamental unit of electric charge is the charge of a single electron, denoted by 'e'. Its value is approximately 1.602 x 10^-19 Coulombs (C). Protons carry a positive charge of equal magnitude, +e.

1.2 Properties of Electric Charge

• Quantization of Charge: Electric charge is always an integer multiple of the elementary charge 'e'. This means that charge cannot exist in arbitrary amounts; it comes in discrete packets. Mathematically, the total charge (Q) on an object is given by Q = n * e, where 'n' is an integer (positive, negative, or zero).
• Conservation of Charge: In any closed system, the total electric charge remains constant. Charge cannot be created or destroyed, only transferred from one object to another or redistributed within an object. For example, when a charged object is brought near a neutral conductor, charges redistribute, but the total charge of the system remains the same.
• Additivity of Charge: The total charge of a system is the algebraic sum of all individual charges present in the system. If a system has charges q₁, q₂, q₃, ..., then the total charge Q = q₁ + q₂ + q₃ + ...

1.3 Methods of Charging Objects

Objects can be charged through three primary methods:

• Charging by Friction (Triboelectric Effect): When two different uncharged objects are rubbed together, electrons are transferred from one object to the other. The object that loses electrons becomes positively charged, and the object that gains electrons becomes negatively charged. The amount of charge acquired by each object is equal in magnitude but opposite in sign.
• Charging by Conduction (Contact): When a charged object touches a neutral conductor, some charge is transferred from the charged object to the conductor. The conductor acquires the same type of charge as the charged object, but its magnitude will be less.
• Charging by Induction: When a charged object is brought near a neutral conductor without touching it, the charges in the conductor redistribute. The side nearer to the charged object develops an opposite charge, and the farther side develops a similar charge. If the conductor is then grounded while the charged object is still nearby, the induced charge of opposite sign is attracted to the charged object and remains on the conductor, while the charge of similar sign is repelled to the ground and escapes. When the charged object is removed, the conductor is left with a net charge opposite to that of the inducing object.

2. Coulomb's Law

Coulomb's Law describes the force of interaction between two point electric charges. It was formulated by the French physicist Charles-Augustin de Coulomb in 1785. The law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

2.1 Statement of Coulomb's Law

The electrostatic force (F) between two point charges, q₁ and q₂, separated by a distance 'r' in a vacuum, is given by:

F ∝ |q₁ * q₂|

F ∝ 1/r²

Combining these, we get:

F = k * (|q₁ * q₂|) / r²

Where 'k' is the constant of proportionality, known as Coulomb's constant.

2.2 Coulomb's Constant (k)

The value of Coulomb's constant 'k' depends on the medium in which the charges are placed. In vacuum (or air, approximately), its value is:

k = 1 / (4π * ε₀) ≈ 9 x 10⁹ N m² C^-2

Here, ε₀ is the permittivity of free space, a fundamental physical constant with a value of approximately 8.854 x 10^-12 C² N^-1 m^-2.

2.3 Nature of the Force

The electrostatic force between two charges can be either attractive or repulsive:

• Repulsive Force: If the two charges have the same sign (both positive or both negative), the force between them is repulsive. Like charges repel.
• Attractive Force: If the two charges have opposite signs (one positive and one negative), the force between them is attractive. Unlike charges attract.

Coulomb's Law is an action-reaction law according to Newton's third law. The force exerted by q₁ on q₂ is equal in magnitude and opposite in direction to the force exerted by q₂ on q₁.

2.4 Vector Form of Coulomb's Law

To describe the force more precisely, we use vector notation. Let r₁₂ be the position vector of charge q₂ with respect to charge q₁, and r₂₁ be the position vector of q₁ with respect to q₂. The unit vector in the direction from q₁ to q₂ is û₁₂ = r₁₂ / |r₁₂|.

The force exerted by charge q₁ on charge q₂ (F₁₂) is:

F₁₂ = k * (q₁ * q₂) / |r₁₂|² * û₁₂

Similarly, the force exerted by charge q₂ on charge q₁ (F₂₁) is:

F₂₁ = k * (q₁ * q₂) / |r₂₁|² * û₂₁

Since û₁₂ = -û₂₁ and |r₁₂| = |r₂₁| = r, we have F₁₂ = -F₂₁, confirming Newton's third law.

2.5 Superposition Principle

When a system contains more than two point charges, the net electrostatic force on any one charge is the vector sum of the electrostatic forces exerted by each of the other charges individually. The presence of other charges does not alter the force between any pair of charges.

If there are 'n' charges q₁, q₂, ..., q_n, the net force on charge q_i (F_i) is:

F_i = F_i1 + F_i2 + ... + F_in = Σ_{j=1, j≠i}ⁿ F_ij

Where F_ij is the force on charge q_i due to charge q_j.

3. Examples and Applications

Coulomb's Law is fundamental to understanding many electrostatic phenomena. Here are a few examples:

3.1 Atomic Structure

The force that holds electrons in orbit around the nucleus of an atom is the electrostatic attraction between the negatively charged electrons and the positively charged nucleus (protons).

3.2 Electrostatic Paint Spraying

In electrostatic painting, paint particles are given a charge. The object to be painted is grounded or given an opposite charge. This causes the charged paint particles to be attracted to the object, leading to a more uniform coating and reduced wastage of paint.

3.3 Xerography (Photocopying)

Photocopiers use electrostatic principles. A charged drum attracts toner particles to form an image, which is then transferred to paper.

3.4 Example Calculation: Force between two point charges

Calculate the force between two point charges, q₁ = +2 μC and q₂ = -3 μC, separated by a distance of 1 meter in vacuum.

Given: q₁ = +2 μC = +2 x 10^-6 C q₂ = -3 μC = -3 x 10^-6 C r = 1 m k = 9 x 10⁹ N m² C^-2

Using Coulomb's Law (magnitude): F = k * (|q₁ * q₂|) / r² F = (9 x 10⁹ N m² C^-2) * (|(+2 x 10^-6 C) * (-3 x 10^-6 C)|) / (1 m)² F = (9 x 10⁹) * (6 x 10^-12) / 1 F = 54 x 10^-3 N F = 0.054 N

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

4. Limitations of Coulomb's Law

Coulomb's Law, as stated, applies strictly to:

• Point Charges: It is valid for charged bodies whose size is negligible compared to the distance separating them. For extended charged bodies, the calculation of force involves integration.
• Stationary Charges: The law describes the force between static charges. If charges are in motion, they produce magnetic fields, and the forces become more complex (electromagnetic forces).
• Vacuum or a Uniform Medium: The value of the constant 'k' (or ε₀) is specific to vacuum. For other media, a relative permittivity (dielectric constant, ε_r) is introduced, modifying the force. The permittivity of the medium is ε = ε₀ * ε_r, and the force becomes F = k * (q₁*q₂) / (ε * r²).

Despite these limitations, Coulomb's Law provides the fundamental basis for understanding electrostatic interactions.