Unit 19: Electric Charges

1. Electric Charges

Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges. The concept of electric charge is fundamental to understanding electricity and magnetism. All matter contains electric charges, and their interactions govern many physical and chemical processes.

Types of Electric Charges

There are two fundamental types of electric charges:

• Positive Charge (+): Associated with protons found in the nucleus of an atom.
• Negative Charge (-): Associated with electrons orbiting the nucleus of an atom.

Objects are typically electrically neutral, meaning they have an equal number of protons and electrons. When an object gains or loses electrons, it becomes charged. If it gains electrons, it becomes negatively charged; if it loses electrons, it becomes positively charged.

A crucial property of electric charges is their interaction:

• Like charges repel: Two positive charges will push each other away, as will two negative charges.
• Unlike charges attract: A positive charge and a negative charge will pull towards each other.

Unit of Electric Charge: The Coulomb (C)

The standard unit of electric charge in the International System of Units (SI) is the Coulomb (C). It is named after the French physicist Charles-Augustin de Coulomb. One Coulomb is defined as the amount of charge that passes a point in one second when an electric current of one ampere is flowing.

In practical applications, charges are often very small, so prefixes like microcoulomb (µC = 10^-6 C) and nanocoulomb (nC = 10^-9 C) are commonly used.

Elementary Charge (e)

The smallest indivisible unit of free electric charge observed in nature is the elementary charge, denoted by e. The magnitude of this charge is:

e = 1.6 x 10-19 C

The charge of a single proton is +e, and the charge of a single electron is -e. All observable charges are integer multiples of this elementary charge.

Quantization of Electric Charge

The principle of quantization of electric charge states that electric charge exists only in discrete packets, rather than in continuous amounts. This means that any charge Q found in nature is always an integer multiple of the elementary charge e.

Mathematically, this is expressed as:

Q = ne

Where:

• Q is the total charge on an object.
• n is an integer (n = ±1, ±2, ±3, ...), representing the number of elementary charges gained or lost.
• e is the elementary charge (1.6 x 10-19 C).

This means you cannot have a charge of, for example, 0.5e or 1.7e. An object can have a charge of -2e (meaning it has two excess electrons) or +5e (meaning it has lost five electrons), but not fractional values.

Example of Quantization:

If an object has a net charge of -4.8 x 10-19 C, how many excess electrons does it have?

Given Q = -4.8 x 10-19 C and e = 1.6 x 10-19 C.

Using the quantization formula Q = ne, we can find n:

n = Q / e n = (-4.8 x 10-19 C) / (-1.6 x 10-19 C) n = 3

Since the charge is negative, the object has 3 excess electrons.

Conservation of Electric Charge

The law of conservation of electric charge is a fundamental principle in physics, stating that the total electric charge in an isolated system remains constant. This means that charge can neither be created nor destroyed, but it can be transferred from one object to another.

When objects are rubbed together (e.g., a glass rod with a silk cloth), electrons are transferred from one object to the other. The glass rod loses electrons and becomes positively charged, while the silk cloth gains those electrons and becomes negatively charged. However, the sum of the charges on the rod and the cloth remains zero (assuming they were initially neutral), demonstrating that the total charge of the isolated system is conserved.

This principle is observed in all known physical processes, from everyday phenomena to nuclear reactions.

2. Charging by Induction

Charging by induction is a method of charging an object without any physical contact between the charging body and the object being charged. This process relies on the redistribution of charges within a conductor due to the proximity of a charged object.

This contrasts with charging by conduction, where charge is transferred through direct physical contact.

The Process of Charging by Induction (using a metal sphere)

Let's illustrate the process of charging a neutral metal sphere positively using a negatively charged rod:

1. Bring a Charged Rod Near the Conductor: Start with an isolated, neutral conducting sphere. Bring a negatively charged rod close to (but not touching) the sphere. The free electrons in the sphere are repelled by the negative rod and move to the side of the sphere farthest from the rod. This leaves an excess of positive charge on the side of the sphere nearest the rod. The sphere is still neutral overall, but its charges are separated (polarized).
2. Ground the Conductor: While the charged rod is still held nearby, connect the sphere to the ground (e.g., by touching it with your finger or connecting it with a wire to the earth). The repelled electrons on the far side of the sphere, being free to move, escape from the sphere and flow into the ground, as the ground acts as an infinite reservoir for charges. The positive charges on the near side remain attracted to the negative rod and cannot move.
3. Remove the Ground Connection: With the charged rod still in place, remove the ground connection. The sphere is now left with an overall net positive charge, as it has lost electrons to the ground. The positive charges are still held on the side near the rod due to attraction.
4. Remove the Charged Rod: Finally, remove the negatively charged rod. With the external inducing charge gone, the excess positive charges on the sphere redistribute themselves uniformly over the entire surface of the sphere due to mutual repulsion. The sphere is now positively charged.

Key outcome: The conductor acquires a charge opposite to that of the inducing charged body. If we had used a positively charged rod, the sphere would have ended up negatively charged.

Applications of Charging by Induction

• Electroscope: An electroscope is a device used to detect the presence and sign of electric charge. It can be charged by induction to determine the polarity of an unknown charge.
• Industrial Applications: Induction charging is used in various industrial processes, such as electrostatic painting (where paint droplets are charged by induction to adhere better to surfaces) and in photocopiers (where toner particles are charged).

3. Coulomb's Law

While we know that like charges repel and unlike charges attract, Coulomb's Law provides a quantitative description of the electrostatic force between two point charges. It was formulated by Charles-Augustin de Coulomb in 1785.

Statement and Formula

"The magnitude of the electric 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."

Mathematically, Coulomb's Law is expressed as:

F = k * |q1 * q2| / r2

Where:

• F is the magnitude of the electrostatic force between the two charges (measured in Newtons, N).
• q1 and q2 are the magnitudes of the two point charges (measured in Coulombs, C). The absolute value signs indicate that we are considering only the magnitude of the charges for calculating the magnitude of the force. The sign of the charges determines the direction.
• r is the distance between the centers of the two point charges (measured in meters, m).
• k is Coulomb's constant, also known as the electrostatic constant.

Coulomb's Constant (k)

The proportionality constant k has a specific value that depends on the medium in which the charges are located. In a vacuum (or approximately in air), its value is:

k = 1 / (4 * pi * ε0)

Where ε0 (epsilon naught) is the permittivity of free space, a fundamental physical constant.

The numerical value of k is approximately:

k ≈ 9 x 109 Nm2/C2

And the value of ε0 is approximately:

ε0 ≈ 8.854 x 10-12 C2/(Nm2)

Direction of the Electric Force

Coulomb's Law describes the magnitude of the force. The direction of the force is determined by the signs of the charges:

• If q1 and q2 have the same sign (both positive or both negative), the force F is repulsive, pushing the charges apart along the line connecting their centers.
• If q1 and q2 have opposite signs (one positive, one negative), the force F is attractive, pulling the charges together along the line connecting their centers.

It is crucial to remember that electric force is a vector quantity, possessing both magnitude and direction.

Comparison with Gravitational Force

Coulomb's Law bears a striking resemblance to Newton's Law of Universal Gravitation (F = Gm1m2/r2). Both are inverse-square laws. However, there are key differences:

Feature	Electric Force (Fe)	Gravitational Force (Fg)
Depends on	Charge (q)	Mass (m)
Nature	Attractive or Repulsive	Always Attractive
Strength	Much stronger	Much weaker
Constant	k (depends on medium)	G (universal constant)

Example Calculation using Coulomb's Law:

Two point charges, q1 = +2.0 µC and q2 = -3.0 µC, are separated by a distance of 0.50 m in a vacuum. Calculate the magnitude and describe the direction of the electrostatic force between them.

Given:

• q1 = +2.0 µC = +2.0 x 10-6 C
• q2 = -3.0 µC = -3.0 x 10-6 C
• r = 0.50 m
• k = 9 x 109 Nm2/C2

Formula:

F = k * |q1 * q2| / r2

Calculation:

F = (9 x 109 Nm2/C2) * |(2.0 x 10-6 C) * (-3.0 x 10-6 C)| / (0.50 m)2 F = (9 x 109) * (6.0 x 10-12) / (0.25) F = (54 x 10-3) / 0.25 F = 0.054 / 0.25 F = 0.216 N

Direction: Since q1 is positive and q2 is negative, the force between them is attractive. Each charge experiences an attractive force of 0.216 N towards the other.

4. Force Between Multiple Electric Charges: The Superposition Principle

In many realistic situations, an object might be subjected to electric forces from more than one other charged object. To determine the net force on a particular charge in such a system, we use the principle of superposition.

The Principle of Superposition

"The total electric force on any charge in a system of charges is the vector sum of the forces exerted on that charge by all the other individual charges. Each individual force is calculated using Coulomb's Law, ignoring the presence of other charges."

This means that if you have a charge Q and it is interacting with charges q1, q2, q3, ... qn, the total force Ftotal on Q is the vector sum of the forces F1, F2, F3, ... Fn, where Fi is the force exerted by qi on Q.

Mathematically:

Ftotal = F1 + F2 + F3 + ... + Fn (vector sum)

Steps for Calculating Total Force

To apply the superposition principle effectively, especially when forces are not collinear, follow these steps:

1. Identify the Target Charge: Determine on which charge you need to calculate the net force.
2. Calculate Individual Forces: For each of the other charges in the system, calculate the magnitude and direction of the force it exerts on the target charge using Coulomb's Law. Treat each pair of charges independently.
3. Resolve Forces into Components: Resolve each individual force vector into its rectangular (x and y) components. Choose a suitable coordinate system (e.g., origin at the target charge or a convenient corner).
4. Sum Components: Add all the x-components together to find the total x-component of the net force (Fx_total = ΣFix). Similarly, sum all the y-components to find the total y-component (Fy_total = ΣFiy).
5. Calculate Resultant Magnitude: Use the Pythagorean theorem to find the magnitude of the total force: Ftotal = sqrt(Fx_total2 + Fy_total2)
6. Calculate Resultant Direction: Determine the direction (angle) of the total force using trigonometry: θ = arctan(Fy_total / Fx_total)

   Remember to consider the quadrant of the resultant vector to get the correct angle.

Detailed Example: Force on a Charge in a System

Consider three point charges arranged in a right triangle as shown below. Calculate the net electric force on charge q1.

• q1 = +1.0 µC located at the origin (0, 0).
• q2 = +2.0 µC located at (0.3 m, 0) on the x-axis.
• q3 = -3.0 µC located at (0, 0.4 m) on the y-axis.

Assume the charges are in a vacuum, so k = 9 x 109 Nm2/C2.

Step 1: Calculate the force exerted by q2 on q1 (F12)

• Distance r12 = 0.3 m.
• Both q1 and q2 are positive, so F12 is repulsive. Since q2 is to the right of q1, F12 acts on q1 in the negative x-direction.
• Magnitude: F12 = k * |q1 * q2| / r122 F12 = (9 x 109) * |(1.0 x 10-6) * (2.0 x 10-6)| / (0.3)2 F12 = (9 x 109) * (2.0 x 10-12) / 0.09 F12 = (18 x 10-3) / 0.09 F12 = 0.2 N
• Vector form: F12 = -0.2 N î

Step 2: Calculate the force exerted by q3 on q1 (F13)

• Distance r13 = 0.4 m.
• q1 is positive and q3 is negative, so F13 is attractive. Since q3 is above q1, F13 acts on q1 in the positive y-direction.
• Magnitude: F13 = k * |q1 * q3| / r132 F13 = (9 x 109) * |(1.0 x 10-6) * (-3.0 x 10-6)| / (0.4)2 F13 = (9 x 109) * (3.0 x 10-12) / 0.16 F13 = (27 x 10-3) / 0.16 F13 = 0.16875 N
• Vector form: F13 = +0.16875 N ĵ

Step 3: Sum the components

• Total force in x-direction: Fx_total = F12x + F13x = -0.2 N + 0 N = -0.2 N
• Total force in y-direction: Fy_total = F12y + F13y = 0 N + 0.16875 N = 0.16875 N

Step 4: Calculate the magnitude of the net force

Ftotal = sqrt(Fx_total2 + Fy_total2) Ftotal = sqrt((-0.2 N)2 + (0.16875 N)2) Ftotal = sqrt(0.04 + 0.0284765625) Ftotal = sqrt(0.0684765625) Ftotal ≈ 0.2617 N

Step 5: Calculate the direction of the net force

θ = arctan(Fy_total / Fx_total) θ = arctan(0.16875 / -0.2) θ = arctan(-0.84375) θ ≈ -40.15°

Since Fx_total is negative and Fy_total is positive, the resultant force vector lies in the second quadrant. The angle -40.15° is relative to the negative x-axis. To express it relative to the positive x-axis, we add 180°:

θ = 180° - 40.15° = 139.85°

Therefore, the net electric force on q1 has a magnitude of approximately 0.2617 N and acts at an angle of approximately 139.85° counter-clockwise from the positive x-axis.