Electric Charges and Fields

Introduction to Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric or magnetic field. There are two types of electric charges:

• Positive Charge (+): Carried by protons.
• Negative Charge (-): Carried by electrons.

Key Properties of Electric Charge:

1. Quantization of Charge: Electric charge exists in discrete packets, and the smallest unit of charge is the charge of an electron or proton. The charge on an electron is e=1.6×10−19e = 1.6 \times 10^{-19}e=1.6×10−19 C.
2. Conservation of Charge: The total charge in an isolated system remains constant. Charge can neither be created nor destroyed but only transferred from one body to another.
3. Additivity of Charge: Charges add algebraically. For example, if two charges of +2e+2e+2e and −3e-3e−3e are combined, the total charge would be −1e-1e−1e.

Coulomb’s Law

Coulomb’s law describes the electrostatic force between two point charges. The force between two charges is:

• Directly proportional to the product of the magnitudes of the two charges.
• Inversely proportional to the square of the distance between them.

This force acts along the line joining the two charges and can be attractive or repulsive depending on the nature of the charges.

Formula:
F is equal to k multiplied by (q1 multiplied by q2) divided by r squared, where

• F is the force between the charges,
• q1 and q2 are the charges,
• r is the distance between them, and
• k is Coulomb’s constant, equal to 9 multiplied by 10 raised to the power of 9 newton meter squared per coulomb squared.

Electric Field

The electric field is a vector quantity that represents the force per unit charge exerted on a small positive test charge placed at any point in space.

Formula:
E is equal to F divided by q, where

• E is the electric field,
• F is the force on the test charge, and
• q is the charge.

Properties of Electric Field:

1. Direction: The electric field produced by a positive charge points radially outward, while that produced by a negative charge points radially inward.
2. Superposition Principle: The net electric field at any point due to a group of charges is the vector sum of the electric fields due to the individual charges.

Electric Field Due to a Point Charge

The electric field due to a point charge Q at a distance r from the charge is given by:

Formula:
E is equal to k multiplied by the absolute value of Q divided by r squared, where

• k is Coulomb’s constant, and
• Q is the point charge.

Electric Field Lines

Electric field lines are imaginary lines used to represent the direction and strength of the electric field.

Properties of Electric Field Lines:

1. They begin at positive charges and end at negative charges.
2. The number of field lines is proportional to the magnitude of the charge.
3. Field lines never intersect.
4. The field is stronger where the lines are closer together.

Electric Flux

Electric flux is the total number of electric field lines passing through a given surface. It gives a measure of how strong the field is across a surface.

Formula:
Electric flux (Phi E) is equal to E multiplied by A, where

• Phi E is the electric flux,
• E is the electric field, and
• A is the area vector perpendicular to the surface.

Gauss’s Law

Gauss’s law relates the electric flux through a closed surface to the charge enclosed by that surface. It is a very useful tool for calculating electric fields for symmetric charge distributions.

Statement of Gauss’s Law:
The total electric flux through a closed surface is equal to the charge enclosed divided by epsilon naught.

Formula:
The integral of E dot dA is equal to Q enclosed divided by epsilon naught, where

• Q enclosed is the charge enclosed by the surface, and
• epsilon naught is the permittivity of free space, equal to 8.85 multiplied by 10 raised to the power of negative 12 coulombs squared per newton meter squared.

Applications of Gauss’s Law

1. Electric Field Due to a Spherical Charge Distribution:
   For a uniformly charged sphere, the electric field inside the sphere is zero, and outside the sphere, it behaves like a point charge.
2. Electric Field Due to an Infinite Line of Charge:
   Gauss’s law can be used to derive the electric field at a distance from an infinitely long line of charge.
3. Electric Field Due to an Infinite Plane Sheet of Charge:
   The electric field due to a uniformly charged infinite plane sheet is constant and does not depend on the distance from the sheet.

Conductors in Electrostatic Equilibrium

Conductors have the following important properties in electrostatic equilibrium:

1. The electric field inside a conductor is zero.
2. Any excess charge on a conductor resides entirely on its surface.
3. The electric field at the surface of a conductor is perpendicular to the surface.
4. The electric potential is constant throughout the conductor and at its surface.

Important Questions for Practice

1. State and explain Coulomb’s law.
2. Define electric field and its properties. How is it different from electric potential?
3. What is electric flux? Derive the expression for electric flux through a plane surface.
4. Explain Gauss’s law and give two applications.
5. How can you find the electric field due to a uniformly charged spherical shell using Gauss’s law?