GATE 2023 PH – Question 35
An electric field is $\mathbf E=\alpha e^{-r^2}\hat r/r$. The flux through a sphere of radius $\sqrt2$ centered at the origin is $\Phi$. Find $\Phi/(2\pi\alpha)$, rounded to two decimal places.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.37 to 0.39
Explanation
The electric field is radial: $\mathbf E=\dfrac{\alpha e^{-r^2}}{r}\hat r$. Its magnitude is the same at every point on a sphere of radius $r$, so the flux through the sphere is simply $E$ times the area:
$$\Phi=E(r)\cdot4\pi r^2=\frac{\alpha e^{-r^2}}{r}\cdot4\pi r^2=4\pi\alpha\,r\,e^{-r^2}.$$
At $r=\sqrt2$:
$$\Phi=4\pi\alpha\sqrt2\,e^{-2}.$$
**Required ratio:**
$$\frac{\Phi}{2\pi\alpha}=2\sqrt2\,e^{-2}=2\times1.4142\times0.13534=\mathbf{0.38}.$$