First slide

Gauss's law and its applications

Question

The electric field in a region is radially outward with magnitude E = $α$ r. If $α$ = 100 V/m² and R = 0.30 m, then the value of charge contained in a sphere of radius R centered at the origin is W x 10^-10 Coulomb. The value of W is

Difficult

Solution

$\begin{aligned} \begin{array}{ll} Given & E = α r \\ When & r = R, E_{R} = α R \\ ∴ & ϕ = E_{R} (area) = α R 4 π R^{2} \end{array} \\ By Gauss theorem, net electric flux \\ = \frac{1}{ε_{0}} \times charge enclosed \\ \begin{aligned} \Rightarrow α R 4 π R^{2} = \frac{1}{ε_{0}} Q_{enclosed} \\ ∴ Q_{enclosed} = (4 π ε_{0}) \cdot α R^{3} \\ Given R = 0.30 m, α = 100 V / m^{2} \\ Q_{enclosed} = \frac{1}{9 \times 10^{9}} \times 100 \times (0.30)^{3} = 3 \times 10^{- 10} C \end{aligned} \end{aligned}$

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