A uniform electric field E = E_xi N/C for x > 0 and E = E _x i N/C for x < 0 are given. A right circular cylinder of length l cm and radius r cm has its centre at the origin and its axis along the x-axis. Find out the net outward flux. Using Gauss's law write the expression for the net charge within the cylinder.

Solution

Given, a right circular cylinder of length l cm and radius r cm along the x- axis.
E = E_xi N/C for x > 0 and,
E = E_x i N/C for x < 0

E and $∆ S$ are parallel for the perimeter area of cylinder.
Therefore, the outer flux is given by,

$φ = E . ∆ S + E . ∆ S = E_{x} (π r^{2}) + E_{x} (π r^{2}) = 2 E_{x} (π r^{2}) N m^{2} C^{- 1}$

Along the curved surface are of the cylinder E and $∆ S$ is perpendicular to each other.
Therefore, the outer flux is 0.

Hence, the net outward flux is given by,
$φ = 2 E_{x} (π r^{2}) = 6.28 E_{x} r^{2} N m^{2} C^{- 1}$

The net charge within the cylinder as per gauss law is given by q $= ε_{\circ} φ$ .

$\Rightarrow 8.854 \times 10^{- 12} \times 3.14 \times E_{x} r^{2}$
$27.8 \times 10^{- 12} E_{x} r^{2}$ C

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