First slide

Gauss's Law

Question

A uniformly charged and infinitely long line having a liner charge density )., is placed at a normal distance y from a point 0. Consider a sphere of radius R with O as centre and R > y. Electric flux through the surface of the sphere is

Difficult

A

Zero
B

$\frac{2 λR}{ε_{0}}$
C

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

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

Solution

$Electric flux, \oint_{S} \vec{E} . d \vec{s} = \frac{q_{in}}{ε_{0}}$
q_in is the charge enclosed by the Gaussian surface which, in the present case, is the surface of given sphere. As shown, length AB of line lies inside the sphere.
$In {ΔOO}^{'} A R^{2} = y^{2} + {(O^{'} A)}^{2} ∴ O^{'} A = \sqrt{R^{2} - y^{2}} and AB = 2 \sqrt{R^{2} - y^{2}} Charge on length AB = 2 \sqrt{R^{2} - y^{2}} \times λ ∴ Electric flux = \oint_{S} \bar{E} . d \vec{s} = \frac{2 λ \sqrt{R^{2} - y^{2}}}{ε_{0}}$

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