GATE 2024 PH – Question 34
$\mathbf E=k(x\hat x+y\hat y)/(x^2+y^2)$. Its flux through a sphere of radius R centered at the origin is $n\pi kR$. The value of n (in integer) is _____
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 4
Explanation
The field is $\mathbf E=\dfrac{k\,(x\hat x+y\hat y)}{x^2+y^2}$. This is a radial field in the $xy$-plane (cylindrical radial), of magnitude $k/\rho$, where $\rho=\sqrt{x^2+y^2}$:
$$\mathbf E=\frac{k}{\rho}\,\hat\rho .$$
**Flux through the sphere of radius $R$.** On the sphere the outward unit normal is $\hat r$, and
$$\hat\rho\cdot\hat r=\frac{\rho}{R}.$$
So
$$\mathbf E\cdot\hat r=\frac{k}{\rho}\cdot\frac\rho R=\frac kR,$$
which is **constant over the whole sphere** (apart from the two poles, a set of measure zero where $\rho=0$).
**Flux:**
$$\Phi=\frac kR\times4\pi R^2=4\pi kR\;\Rightarrow\;n=\mathbf{4}.$$