Question

State Gauss’s theorem. Using this theorem, derive electric field intensity due to thin uniformly charged infinite plane sheet.

Hint:

The electric field due to an infinite plane sheet with uniform charge density is constant and does not depend on the distance from the sheet.

Answer 1

Step 1: Gauss’s Theorem.
Gauss's theorem (or Gauss's law) states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, Gauss’s law is given by: \[ \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{\text{enc}}}{\epsilon_0} \] where: - \( \oint \vec{E} \cdot d\vec{A} \) is the electric flux through a closed surface, - \( Q_{\text{enc}} \) is the total charge enclosed within the surface, - \( \epsilon_0 \) is the permittivity of free space.
Step 2: Application of Gauss’s Law to an Infinite Plane Sheet.
Consider an infinite plane sheet with uniform charge density \( \sigma \) (charge per unit area). To find the electric field due to this sheet, we apply Gauss's law using a Gaussian surface in the form of a rectangular box (Gaussian pillbox) that is symmetrically placed around the plane. The box should have one face above and one face below the plane sheet.
Step 3: Symmetry and Field Direction.
By symmetry, the electric field will be perpendicular to the surface of the sheet and will have the same magnitude at equal distances on both sides of the sheet. Let the electric field be \( E \) on both sides of the sheet, directed away from the surface if the charge is positive.
Step 4: Electric Flux Calculation.
The total flux through the Gaussian surface is the sum of the flux through the top and bottom faces. The area of each face is \( A \), and the flux is: \[ \Phi_E = E \cdot A + E \cdot A = 2EA \] Since the charge enclosed is \( Q_{\text{enc}} = \sigma A \), Gauss’s law gives: \[ 2EA = \dfrac{\sigma A}{\epsilon_0} \] Simplifying, we get the electric field: \[ E = \dfrac{\sigma}{2 \epsilon_0} \] Step 5: Conclusion.
The electric field intensity due to an infinitely charged plane sheet is: \[ E = \dfrac{\sigma}{2 \epsilon_0} \] where \( \sigma \) is the surface charge density, and \( \epsilon_0 \) is the permittivity of free space.