First slide

Electric flux

Question

A uniformly charged and infinitely long line having a liner charge density $λ$ is placed at a normal distance y from a point O. Consider a sphere of radius R with O as the center and R > y. Electric flux through the surface of the sphere is

Moderate

A

zero
B

$\frac{2 λ R}{ε_{0}}$
C

$\frac{2 λ \sqrt{R^{2} - y^{2}}}{ε_{0}}$
D

$\frac{λ \sqrt{R^{2} + y^{2}}}{ε_{0}}$

Solution

$Electric flux ∮_{S} \vec{E} \cdot \vec{d} s = \frac{q_{in}}{ε_{0}}$
q_in, is the charge enclosed by the Gaussian surface, which, in the present case, is the surface of the given sphere. As shown, length AB of line lies inside the sphere. In $Δ O O^{'} A$ ,
$\begin{array}{l} R^{2} = Y^{2} + {(O^{'} A)}^{2} \\ ∴ O^{'} A = \sqrt{R^{2} - y^{2}} \\ and A B = 2 \sqrt{R^{2} - y^{2}} \end{array}$
$Charge on length A B is 2 \sqrt{R^{2} - y^{2}} \times λ$
Therefore, electric flux is
$∮_{S} \vec{E} \cdot \vec{d} s = \frac{2 λ \sqrt{R^{2} - y^{2}}}{ε_{0}}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App