ODISHA CLASS 9

Definition: Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric and magnetic field. There are two types of charges: positive (+) and negative (-).
Unit: The SI unit of charge is the coulomb (C).

Principle: The total charge in an isolated system remains constant over time. Charge cannot be created or destroyed, only transferred between bodies.
Implication: This principle is a fundamental law of nature, ensuring that charge is conserved in all processes.

Statement: Coulomb's law gives the force between two point charges. It states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Mathematically:
$F = \frac{k_e |q_1 q_2|}{r^2}$
where:
- $F$ is the magnitude of the electrostatic force,
- $k_e = 9 \times 10^9 \, \text{N m}^2 \text{C}^{-2}$ (Coulomb's constant),
- $q_1, q_2$ are the charges,
- $r$ is the distance between the charges.
The force is attractive if the charges are opposite, and repulsive if the charges are the same.

Superposition Principle: The total force on any charge due to a group of other charges is the vector sum of the individual forces exerted on it by each of the other charges.

If there are multiple charges $q_1, q_2, q_3, \dots$ , the total force on charge $q_1$ is:
$\vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \dots$
where $\vec{F}_{12}$ , $\vec{F}_{13}$ , etc., are the individual forces on $q_1$ due to other charges.

When charge is spread over a continuous region (e.g., a charged rod, surface, or volume), we can find the electric field or force by integrating the contribution from each infinitesimal charge element.

For example, for a linear charge distribution, the charge element $dq$ is considered along the length of the wire, and the electric field is calculated by integrating the contributions from all such elements.

Definition: The electric field ( $\vec{E}$ ) is a vector field that represents the force per unit charge experienced by a test charge placed at a point in space.
$\vec{E} = \frac{F}{q}$
where $F$ is the force on a test charge $q$ .
Units: The SI unit of electric field is N/C (Newtons per Coulomb).
Electric Field Due to a Point Charge: For a point charge $q$ , the electric field at a distance $r$ from the charge is given by:
$E = \frac{k_e |q|}{r^2}$
The direction of the electric field depends on the sign of the charge. It points away from positive charges and toward negative charges.

Electric Field Lines: These are imaginary lines used to represent the direction and strength of an electric field. They point away from positive charges and toward negative charges.
- The number of lines represents the strength of the electric field.
- Electric field lines never cross each other.
- The closer the lines, the stronger the field.

Definition: An electric dipole consists of two equal and opposite charges, $+q$ and $-q$ , separated by a distance $d$ . It has a dipole moment, defined as:
$\vec{p} = q \cdot \vec{d}$
where $\vec{d}$ is the vector from the negative to the positive charge.
Electric Field Due to a Dipole: The electric field at a point due to an electric dipole depends on the distance and orientation of the point relative to the dipole. In the case of a point on the axis of the dipole:
$E = \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^3}$
where $p$ is the dipole moment and $r$ is the distance from the center of the dipole.

When an electric dipole is placed in a uniform electric field $\vec{E}$ , it experiences a torque that tends to align the dipole moment $\vec{p}$ with the electric field. The torque is given by:
$\tau = pE \sin \theta$
where $\theta$ is the angle between the dipole moment and the electric field.

Definition: Electric flux ( $\Phi_E$ ) is the total electric field passing through a given area. It is defined as the product of the electric field and the area through which it passes, and the cosine of the angle between them:
$\Phi_E = E \cdot A \cdot \cos \theta$

Gauss's Law states that the electric flux through a closed surface is proportional to the enclosed electric charge. Mathematically:
$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}$
where $Q_{\text{enc}}$ is the total charge enclosed within the surface, and $\epsilon_0$ is the permittivity of free space.
Applications of Gauss's Law:

a. Field Due to an Infinitely Long Straight Wire:
The electric field at a distance $r$ from a long, uniformly charged wire is:
$E = \frac{2 k_e \lambda}{r}$
where $\lambda$ is the linear charge density.

b. Field Due to a Uniformly Charged Infinite Plane Sheet:
The electric field due to an infinitely charged plane sheet is constant and given by:
$E = \frac{\sigma}{2 \epsilon_0}$
where $\sigma$ is the surface charge density.

c. Field Due to a Uniformly Charged Thin Spherical Shell:
- Outside the shell ( $r > R$ ): The electric field behaves as if the entire charge is concentrated at the center:
  $E = \frac{k_e Q}{r^2}$
- Inside the shell ( $r < R$ ): The electric field is zero:
  $E = 0$