Question

State Gauss’ theorem in electrostatics.
Apply this theorem to derive an expression for the electric intensity due to an infinite sheet of charge and hence deduce the expression for electric intensity due to two parallel oppositely charged thin infinite sheets of charge at a point

(ii) Outside the sheets.

Accepted Answer

Gauss's theorem:

According to this theorem the " the net electric flux through any closed surface is numerically equal to $\frac{1}{ε_{0}}$ times the charge enclosed by the surface."

Mathematically: $ϕ = \oint \vec{E} \cdot \vec{d A}$ $= \frac{1}{ε_{0}} \cdot q = \frac{q_{i n}}{ϵ_{0}}$

Expression for electric field intensity at any point due to infinite plane sheet of charge

Let a infinite plane sheet of charge having surface charge density $σ$

as $σ = \frac{q}{A};$ Where A = plate area

At any point outside this sheet electric field intensity is to be determined. For this purpose a Gaussian cylindrical is considered So, that $\vec{E}$ becomes perpendicular to the plane, so that magnitude become same in both sides of the sheet.

Now total electric flux through both sides of the sheet $= ϕ = \oint \vec{E} \cdot \vec{d A} + \oint \vec{E} \cdot \vec{d A}$

$= \oint E d A \cos 0^{\circ} + \oint E d A \cos 0^{\circ}$

$= E \oint d A + E \oint d A$

$= 2 E A . . . . . . . . (i)$

Net charge enclosed by the Gaussian cylinder be q.

So according to Gauss's theorem electric flux $= ϕ = \frac{q_{i n}}{ϵ_{0}} = \frac{q}{ϵ_{0}}$ .......(ii)

From equation (i) and (ii), we get $2 E A = \frac{q}{ϵ_{0}}$

(ii) Electric field outside the sheet.

When point is considered outside the plates then $E_{n e t_{}}$ left to the plates

$= E_{2} - E_{1} = \frac{σ_{2}}{2 E_{0}} - \frac{σ_{1}}{2 ϵ_{0}}$ $= \frac{1}{2 ϵ_{0}} (σ_{2} - σ_{1})$ , towards the larger magnitude of $σ$

And right to the plates

$= E_{n e t} = E_{1} - E_{2} = \frac{σ_{1}}{2 ϵ_{0}} - \frac{6_{2}}{2 ϵ_{0}} = \frac{1}{2 ε_{0}} (σ_{1} - σ_{2})$ towards the larger magnitude of $σ$

if $σ_{1} = σ_{2} = σ then, E_{n e t} =0$

Important Questions on Gauss' Theorem

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