Name: Electric and Gravitational Forces
Uploaded: 2019-07-19
Duration: 3 min 11 s
Description: Do you know the difference and similarities between the gravitational force and electrostatic force? Watch this video to explore it. Don't miss it.

Question

State Gauss's theorem and express it mathematically. Using it, derive an expression for the electric field intensity at a point near a thin infinite plane sheet of charge density

σ {Cm}^{- 2}

.

Accepted Answer

Gauss’s theorem gives the relationship between the total flux passing through any closed surface and the net charge enclosed within the surface.

It states that the total flux through a closed surface is $\frac{1}{ε_{0}}$ times the net charge enclosed by the closed surface.

Mathematically, it can be expressed as

$ϕ_{E} = \oint \vec{E} \cdot d \vec{S} = \frac{q}{ε_{0}}$

Consider a thin infinite plane sheet of charge with uniform surface charge density $σ$ .

By symmetry, electric field $E$ points outwards normal to the sheet. Also, it must have same magnitude and opposite direction at two points $P$ and $P'$ equidistant from the sheet and on opposite sides.

We choose cylindrical Gaussian surface of cross-sectional area $A$ and length $2 r$ with its axis perpendicular to the sheet.

As the lines of force are parallel to the curved surface of the cylinder, the flux through the curved surface is zero.

The flux through the plane-end faces of the cylinder is $ϕ_{E} = E A + E A = 2 E A$

Charge enclosed by the Gaussian surface, $q = σ A$

According to Gauss’s theorem,

$ϕ_{L} = 2 E A = \frac{q}{ε_{0}} = \frac{σ A}{ε_{0}}$

$\Rightarrow E = \frac{σ}{2 ε_{0}}$

In vector form it can be expressed as $\vec{E} = \frac{σ}{2 ε_{o}} \hat{n}$

Where $\hat{n}$ is the unit vector normal to the sheet.

State Gauss's theorem and express it mathematically. Using it, derive an expression for the electric field intensity at a point near a thin infinite plane sheet of charge density $σ {Cm}^{- 2}$ .

Important Questions on Electric Charges and Field

Learn and Prepare for any exam you want !

State Gauss's theorem and express it mathematically. Using it, derive an expression for the electric field intensity at a point near a thin infinite plane sheet of charge density σ Cm−2.

Important Questions on Electric Charges and Field

Learn and Prepare for any exam you want !

State Gauss's theorem and express it mathematically. Using it, derive an expression for the electric field intensity at a point near a thin infinite plane sheet of charge density $σ {Cm}^{- 2}$ .