Electric Charges and Fields (CLASS-12 NOTES) - Key Concepts, and Formulas

Electric Charges and Fields - Electric charges and fields form the cornerstone of electrostatics, a branch of physics that deals with stationary electric charges and their interactions. This comprehensive article delves into the fundamental concepts, principles, and equations of electric charges and fields, providing insights into Coulomb's law, electric field, electric flux, and Gauss's theorem.

Basics of Electric Charges

What are Electric Charges?

• Definition: Electric charge is a fundamental property of matter that causes it to experience a force in the presence of other charges.
• Types of Charges:
  • Positive Charge: Due to a deficiency of electrons.
  • Negative Charge: Due to an excess of electrons.

Properties of Electric Charges

1. Quantization:
   • Charge exists in discrete packets.
   • Smallest unit of charge: $e = 1.6 \times 10^{-19} \, \text{C}$.
   • Formula: $q = n \cdot e$, where $n$ is an integer.
2. Conservation of Charge:
   • The total charge in an isolated system remains constant.
3. Attraction and Repulsion:
   • Like charges repel, unlike charges attract.

Charging Methods

1. Charging by Rubbing:
   • Example: Rubbing a glass rod with silk.
2. Charging by Induction:
   • A charged object induces a charge in a nearby neutral object without direct contact.
3. Charging by Contact:
   • Direct transfer of charge from one object to another through physical contact.

Coulomb’s Law

Statement

Coulomb’s law quantifies the force between two point charges:

$$F = k_e \frac{q_1 q_2}{r^2},$$

where:

• $F$: Electrostatic force,
• $q_1$ and $q_2$: Charges,
• $r$: Distance between the charges,
• $k_e = \frac{1}{4 \pi \epsilon_0}$: Coulomb’s constant.

Permittivity of Medium

• In a vacuum: $k_e = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2$.
• Relative permittivity: $\epsilon_r = \frac{\epsilon}{\epsilon_0}$.

Superposition Principle

• The net force on a charge due to multiple other charges is the vector sum of all individual forces.

Electric Field

Definition

The electric field ($E$) is the region around a charge where another charge experiences a force:

$$E = \frac{F}{q},$$

where $F$ is the force experienced by a test charge $q$.

Electric Field Due to Point Charges

1. Single Point Charge: $$E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2}.$$
2. Electric Dipole:
   • On the axial line: $$E_{\text{axial}} = \frac{2p}{4 \pi \epsilon_0 r^3}.$$
   • On the equatorial line: $$E_{\text{equatorial}} = \frac{p}{4 \pi \epsilon_0 r^3}.$$

Electric Field Lines

• Imaginary lines representing the direction of the electric field.
• Properties:
  1. Lines originate from positive charges and terminate at negative charges.
  2. Lines never intersect.
  3. Denser lines indicate stronger fields.

Electric Flux and Gauss's Theorem

Electric Flux ($\Phi_E$)

• The total number of electric field lines passing through a surface: $$\Phi_E = \vec{E} \cdot \vec{A} = E \cdot A \cdot \cos \theta.$$
• SI Unit: $\text{Nm}^2/\text{C}$.

Gauss's Theorem

• States that the total electric flux through a closed surface is proportional to the enclosed charge: $$\oint \vec{E} \cdot \vec{dA} = \frac{q_{\text{enclosed}}}{\epsilon_0}.$$

Applications of Gauss’s Theorem

1. Electric Field Due to a Point Charge: $$E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2}.$$
2. Infinite Line Charge: $$E = \frac{\lambda}{2 \pi \epsilon_0 r}.$$
3. Spherical Shell:
   • Outside the shell: $$E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2}.$$
   • Inside the shell: $$E = 0.$$

Electric Potential

Definition

Electric potential ($V$) at a point is the work done in bringing a unit positive charge from infinity to that point:

$$V = \frac{W}{q}.$$

Potential Due to Point Charge

$$V = \frac{1}{4 \pi \epsilon_0} \frac{q}{r}.$$

Potential Energy

• Potential energy of a system of charges: $$U = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r}.$$

Dipole in an Electric Field

Dipole Moment ($p$)

• Defined as: $$p = q \cdot 2a,$$ where $q$ is the charge and $2a$ is the distance between charges.

Torque on a Dipole

• Torque experienced by a dipole in a uniform electric field: $$\tau = pE \sin \theta.$$

Potential Energy of a Dipole

$$U = -pE \cos \theta.$$

Important Formulas and Equations

1. Coulomb's Force: $$F = k_e \frac{q_1 q_2}{r^2}.$$
2. Electric Field: $$E = \frac{F}{q}.$$
3. Electric Flux: $$\Phi_E = \vec{E} \cdot \vec{A}.$$
4. Gauss’s Theorem: $$\oint \vec{E} \cdot \vec{dA} = \frac{q_{\text{enclosed}}}{\epsilon_0}.$$
5. Dipole Moment: $$p = q \cdot 2a.$$
6. Torque on a Dipole: $$\tau = pE \sin \theta.$$

FAQs About Electric Charges and Fields

1. What is the significance of Coulomb's law?

Coulomb's law quantifies the electrostatic force between two charges and helps explain phenomena such as attraction, repulsion, and charge distribution.

2. How is electric flux calculated?

Electric flux is the product of the electric field, the area through which it passes, and the cosine of the angle between them.

3. What is Gauss's theorem used for?

Gauss's theorem simplifies the calculation of electric fields for symmetric charge distributions, such as spherical or cylindrical systems.

4. How does an electric dipole behave in a uniform field?

An electric dipole experiences torque in a uniform electric field, aligning itself with the field direction.

5. Why are electric field lines absent inside a conductor?

Inside a conductor, charges redistribute themselves to cancel any internal electric field.