Electric Charges and Conservation of Charge

Understanding electric charges is fundamental to comprehending electricity and magnetism. Charges are a property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative.

The basic unit of electric charge is the charge of a single electron, which is negative, or a single proton, which is positive. The symbol for charge is 'q' or 'Q'. The SI unit of electric charge is the Coulomb (C). The charge of an electron is approximately -1.602 x 10^-19 C, and the charge of a proton is approximately +1.602 x 10^-19 C.

Matter is generally made up of atoms, which consist of protons (positive charge), neutrons (no charge), and electrons (negative charge). In a neutral atom, the number of protons equals the number of electrons, resulting in a net charge of zero.

How Objects Acquire Charge

Objects can become electrically charged through three primary mechanisms:

1. Conduction: This occurs when a charged object touches a neutral object. Charge is transferred from the charged object to the neutral object, causing both to become charged. For example, if a negatively charged rod touches a metal sphere, some electrons will move from the rod to the sphere, leaving both negatively charged.
2. Friction: When two different materials are rubbed together, electrons can be transferred from one material to the other. The material that loses electrons becomes positively charged, and the material that gains electrons becomes negatively charged. This is known as charging by rubbing. For instance, rubbing a glass rod with silk causes the rod to become positively charged and the silk to become negatively charged.
3. Induction: This method involves charging an object without direct contact. If a charged object is brought near a neutral conductor, it can cause a separation of charge within the conductor. If the conductor is then grounded, charge can flow to or from the ground, leaving the conductor with a net charge opposite to that of the inducing object.

Properties of Electric Charge

Electric charges exhibit several key properties:

• Like charges repel, and unlike charges attract. Two positive charges will push each other away, two negative charges will push each other away, but a positive and a negative charge will pull towards each other.
• Charge is quantized. Electric charge always exists in discrete units that are multiples of the elementary charge 'e'. This means you cannot have a fraction of an electron's charge. The charge 'q' on any object is given by q = n * e, where 'n' is an integer (positive or negative).
• Charge is conserved. This is a fundamental principle in physics.

Conservation of Charge

The principle of conservation of charge states that the total electric charge in an isolated system remains constant over time. Charge cannot be created or destroyed; it can only be transferred from one object to another or redistributed within an object.

Consider a system of two isolated objects. If object A has a charge of +5 C and object B has a charge of -3 C, the total charge of the system is +5 C + (-3 C) = +2 C. If these objects interact and exchange charge, the total charge of the system will still be +2 C, even if the individual charges on A and B change.

A classic example of charge conservation is in nuclear reactions. For instance, in beta decay, a neutron decays into a proton, an electron, and an antineutrino.

The conservation of charge is a universal law and applies to all known physical processes. This principle is crucial for understanding and balancing charge in various electrical and nuclear phenomena.

Coulomb's Law for Point Charges

Coulomb's Law describes the electrostatic force between two stationary point charges. A point charge is an idealized model of a charged particle whose size is negligible compared to the distance separating it from other charges. This law was formulated by 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. The force acts along the line joining the two charges.

The Formula

Mathematically, Coulomb's Law is expressed as:

F = k * |q₁q₂| / r²

Where:

• F is the magnitude of the electrostatic force between the two charges.
• q₁ and q₂ are the magnitudes of the two point charges.
• r is the distance between the centers of the two point charges.
• k is Coulomb's constant, also known as the electrostatic constant.

The direction of the force depends on the signs of the charges:

• If q₁ and q₂ have the same sign (both positive or both negative), the force is repulsive.
• If q₁ and q₂ have opposite signs (one positive and one negative), the force is attractive.

Coulomb's Constant (k)

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

k ≈ 8.9875 x 10⁹ N m²/C²

This value is often approximated as 9 x 10⁹ N m²/C² for simpler calculations.

The constant 'k' can also be expressed in terms of the permittivity of free space, denoted by ε₀:

k = 1 / (4πε₀)

The value of ε₀ is approximately 8.854 x 10^-12 C²/(N m²).

Vector Form of Coulomb's Law

To fully describe the force, we use vector notation. Let r₁₂ be the position vector from charge q₁ to charge q₂. The force exerted by q₁ on q₂, denoted as F₁₂, is given by:

F₁₂ = k * (q₁q₂ / r²) * r̂₁₂

Where r̂₁₂ is a unit vector in the direction from q₁ to q₂. If the charges are repulsive, F₁₂ points away from q₁. If they are attractive, F₁₂ points towards q₁.

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

F₂₁ = k * (q₁q₂ / r²) * r̂₂₁

Note that r̂₂₁ = -r̂₁₂, and by Newton's third law, F₁₂ = -F₂₁, meaning the forces are equal in magnitude and opposite in direction.

Example Calculation

Let's calculate the force between two point charges: q₁ = +2 μC and q₂ = -3 μC, separated by a distance of 10 cm in a vacuum.

First, convert units:

• q₁ = +2 x 10^-6 C
• q₂ = -3 x 10^-6 C
• r = 10 cm = 0.1 m

Using Coulomb's Law (magnitude):

F = (9 x 10⁹ N m²/C²) * |(+2 x 10^-6 C) * (-3 x 10^-6 C)| / (0.1 m)²

F = (9 x 10⁹) * | -6 x 10^-12 | / 0.01

F = (9 x 10⁹) * (6 x 10^-12) / 0.01

F = 54 x 10^-3 / 0.01

F = 0.054 / 0.01

F = 5.4 N

Since the charges are opposite in sign, the force is attractive.

Superposition Principle for Electric Charges

In reality, systems often involve more than two charges. The superposition principle allows us to calculate the net electrostatic force on a particular charge due to multiple other charges. It states that the total force on a charge is the vector sum of the forces exerted by each of the other individual charges acting alone.

This principle is valid because the electrostatic force between any two charges depends only on those two charges and their separation, and is not affected by the presence of other charges.

Applying the Principle

Consider a system of 'n' point charges q₁, q₂, ..., q_n. To find the net force F₁ on charge q₁, we calculate the force exerted by each other charge (q₂, q₃, ..., q_n) on q₁ individually, as if the other charges were absent. Then, we add these individual forces as vectors.

The net force F₁ on charge q₁ is given by:

F₁ = F₂₁ + F₃₁ + F₄₁ + ... + F_n1

Using Coulomb's Law for each term:

F₁ = Σ_j=2ⁿ F_j1

F₁ = Σ_j=2ⁿ [ k * (q_jq₁ / r_j1²) * r̂_j1 ]

Here, F_j1 is the force exerted by charge q_j on charge q₁, r_j1 is the distance between q_j and q₁, and r̂_j1 is the unit vector pointing from q_j to q₁.

Example: Force on a Charge in a Triangle

Suppose we have three charges: q₁ = +5 μC at the origin (0,0), q₂ = -2 μC at (1m, 0), and q₃ = +3 μC at (0, 1m). We want to find the net force on q₁.

Step 1: Calculate the force F₂₁ exerted by q₂ on q₁.

• Distance r₂₁ = 1 m.
• Unit vector r̂₂₁ points from q₂ to q₁, which is in the negative x-direction. So, r̂₂₁ = -i.
• F₂₁ = k * (q₂q₁ / r₂₁²) * r̂₂₁
• F₂₁ = (9 x 10⁹) * ((-2 x 10^-6) * (5 x 10^-6) / 1²) * (-i)
• F₂₁ = (9 x 10⁹) * (-10 x 10^-12) * (-i)
• F₂₁ = (9 x 10⁹) * (-10 x 10^-12) * (-1) i
• F₂₁ = 90 x 10^-3 i = 0.09 i N

This force is attractive (since charges are opposite) and points from q₁ towards q₂, which is the positive x-direction. Wait, the unit vector is from q2 to q1, so it points left. The force is attractive, so it pulls q1 towards q2. So it should be in the +x direction. Let's recheck the unit vector direction. r_21 is the vector from 2 to 1. If 2 is at (1,0) and 1 is at (0,0), the vector from 2 to 1 is (0-1, 0-0) = (-1, 0). The unit vector is (-1, 0) / |(-1,0)| = (-1,0). So r_hat_21 is -i. The force F_21 is k * q2*q1 / r^2 * r_hat_21. F_21 = k * (-2*5*10^-12) / 1^2 * (-i) = k * (-10*10^-12) * (-i) = k * 10*10^-12 * i. So F_21 = 9 * 10^9 * 10 * 10^-12 * i = 90 * 10^-3 * i = 0.09i N. This is correct.

Step 2: Calculate the force F₃₁ exerted by q₃ on q₁.

• Distance r₃₁ = 1 m.
• Unit vector r̂₃₁ points from q₃ to q₁, which is in the negative y-direction. So, r̂₃₁ = -j.
• F₃₁ = k * (q₃q₁ / r₃₁²) * r̂₃₁
• F₃₁ = (9 x 10⁹) * ((3 x 10^-6) * (5 x 10^-6) / 1²) * (-j)
• F₃₁ = (9 x 10⁹) * (15 x 10^-12) * (-j)
• F₃₁ = 135 x 10^-3 * (-j) = -0.135 j N

This force is repulsive (since charges are positive) and points from q₁ away from q₃, which is the negative y-direction.

Step 3: Find the net force F₁.

• F₁ = F₂₁ + F₃₁
• F₁ = (0.09 i) + (-0.135 j) N
• F₁ = (0.09 i - 0.135 j) N

The magnitude of the net force is |F₁| = sqrt((0.09)² + (-0.135)²) = sqrt(0.0081 + 0.018225) = sqrt(0.026325) ≈ 0.162 N.

The direction can be found using the arctangent of the y-component divided by the x-component.