Question

A cylinder of radius R and length L is placed in a uniform electric field E parallel to the cylinder axis. The total flux for the surface of the cylinder is given by :

• A. $2\pi R^{2}E$
• B. $\frac{\pi R^2}{E}$
• C. $\frac{R}{E}$
• D. Zero

Answer 1

The correct option is D

Zero

Step 1: Given data

Radius of the cylinder$=R$

Length of the cylinder$=L$

Strength of the electric field$=E$

Step 2: Assumptions

Flux through surface “A”$={\varphi }_A$

Flux through surface “B”$={\varphi }_B$

Flux through curved surface “C”$={\varphi }_c$

Total flux for the surface of the cylinder$={\varphi }_t$

Small elementary area over the surface “C”$=ds$

Step 3: Formula used

${\varphi }_t={\varphi }_A+{\varphi }_B+{\varphi }_c$ ………………………..(a)

Step 3: Calculation of the total flux through the surface of the cylinder

Flux through the surface “A”, ${\varphi }_A=E\times \pi R^2$ …………………………………..(b)

Logically speaking, the flux through the surface “B” will be opposite and equal to the flux through surface “B”

Flux through the surface, “B”, ${\varphi }_B=-E\times \pi R^2$…………………………………(c)

Flux through curved surface “C”, ${\varphi }_c=\int E.ds$

${\varphi }_c=\int Eds\cos {90}^0$

${\varphi }_c=0$………………………………..(d)

Using equations (b), (c) and (d) in equation (a), we get

$$\begin{gathered} {\varphi }_t=(E\times \pi R^2)-\left(E\times \pi R^2\right)+0 \\ {\varphi }_t=o \end{gathered}$$

Hence, option (D) is correct.