The electric field in a certain region is acting radially outward and is given by \( E = Ar^{2} \). A charge contained in a sphere of radius 'a' centred at the origin of the field, will be given by
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 4\pi\varepsilon_0 A a^{4} \)
Apply Gauss's law to the sphere of radius \( a \). The field is radial and has the same size at every point of the sphere, so the flux is the field times the surface area.
\( \displaystyle \Phi = E(a) \times 4\pi a^{2} = A a^{2} \times 4\pi a^{2} = 4\pi A a^{4} \)
\( \displaystyle q = \varepsilon_0\,\Phi = 4\pi\varepsilon_0 A a^{4} \)