Question

If the constant electric field in the figure has a magnitude $E_0$, calculate the total electric flux through the paraboloidal surface.

• A. $E_0\pi R^2$
• B. $2E_0\pi R^2$
• C. $2E_0\pi R$
• D. $\frac{E_0}{2\pi R}$

Answer 1

The correct option is A $E_0\pi R^2$

Since number of field lines entering through circular surface is equal to number of field lines leaving the paraboloid surface.

So, the total flux through the closed surface is zero.

We have,
Flux through plane circular part $\left({\varphi }_c\right)+$ Flux through parabolidal surface$\left({\varphi }_p\right)=0$

$\Rightarrow {\varphi }_c+{\varphi }_p=0$

$\Rightarrow {\varphi }_p=-{\varphi }_c...\left(1\right)$

Now, the ${\varphi }_c$ can be calculated by ,

${\varphi }_c=\left(E_0\hat{i}\right).\pi R^2(-\hat{i})$

$\Rightarrow {\varphi }_c=E_0\pi R^2...\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, we have

${\varphi }_p=E_0\pi R^2$

Hence, option (a) is correct answer.