Question

Derive an expression for Coulomb's law of electrostatics by Gauss's law.
OR
What is capacity of a conductor? Write the factors affecting the capacity of a conductor.

Hint:

The capacitance increases when the surface area of the conductor is larger or when the distance between conductors is smaller. A dielectric material between the conductors increases the capacitance.

Answer 1

a. Deriving Coulomb's Law from Gauss's Law:
Gauss's law states that the electric flux \( \Phi_E \) through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is expressed as:
\[ \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]
Where:
- \( \vec{E} \) is the electric field,
- \( d\vec{A} \) is the differential area element on the closed surface,
- \( Q_{\text{enc}} \) is the charge enclosed within the surface,
- \( \epsilon_0 \) is the permittivity of free space.
For a point charge \( Q \), we use a spherical Gaussian surface of radius \( r \) centered at the point charge. The electric field \( \vec{E} \) is radially symmetric and has the same magnitude at every point on the surface. The surface area of a sphere is \( 4\pi r^2 \), so the electric flux is:
\[ \Phi_E = E \times 4\pi r^2 \] Using Gauss's law:
\[ E \times 4\pi r^2 = \frac{Q}{\epsilon_0} \] Solving for the electric field \( E \):
\[ E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2} \] This is Coulomb's law, which states that the electric field due to a point charge is inversely proportional to the square of the distance from the charge and directly proportional to the magnitude of the charge.
b. Capacity of a Conductor:
The capacity of a conductor refers to the amount of charge a conductor can hold at a particular potential. It is also known as the electrostatic capacitance of the conductor. The unit of capacitance is the farad (F), and it is defined as the amount of charge per unit potential:
\[ C = \frac{Q}{V} \] Where:
- \( C \) is the capacitance,
- \( Q \) is the charge stored on the conductor,
- \( V \) is the potential of the conductor.
Factors Affecting the Capacity of a Conductor:
The following factors affect the capacitance of a conductor:
1. Size of the Conductor: The larger the size of the conductor (particularly the surface area), the higher the capacitance, as it can store more charge at the same potential.
2. Distance Between Conductors: The closer the conductors are to each other, the higher the capacitance, because the electric field between the conductors is stronger.
3. Dielectric Material: The material between the conductors affects the capacitance. A dielectric material with a higher relative permittivity increases the capacitance by reducing the electric field between the conductors, allowing more charge to be stored.
4. Shape of the Conductor: The shape of the conductor can also affect the capacitance. For example, the capacitance of a parallel plate capacitor depends on the area of the plates and the distance between them.