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My question is from Physics For Scientists And Engineers 7th:

A particle with charge Q is located immediately above the center of the flat face of a hemisphere of radius R, as shown in the figure.

  • (a) What is the electric flux through the flat surface,
  • (b) through the curved surface?

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What is the electric flux if $𝛿≠0$, for example $2R$? If it is the same, then how we can prove this? And how we can calculate it?

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Since we don't answer homework-type questions, I'll try to give some hints.

  • What would be the flux through the surface of the sphere, if it was a full and not a semi-sphere?
  • Which of Maxwell's equations could we use?
  • Can we deduct the flux through the semi-sphere from that?
  • Can we use the same equation to answer the second part of the question?

After some clarification I think a complete answer would be instructional.

For the flux through the flat surface the most direct approach would be to simply calculate the integral of the electric field, which is known.

$$ \Phi = \iint \vec{E} d\vec{A} = \iint \vec{E} \vec{n} \, dA = \int_0^{2\pi} d\phi \int_0^R r\,dr \, E\cos{\theta} = 2\pi \int_0^R r\,dr \, E\cos{\theta}$$

The magnitude of the electric field at the surface is $$ E = \frac{Q}{4\pi\epsilon_0 (\delta^2 + r^2)} $$ and by trigonometry $$ \cos(\theta) = \frac{\delta}{\sqrt{r^2+\delta^2}}$$

So $$ \Phi = 2\pi\int_0^R \frac{Q r \delta}{4\pi\epsilon_0 (r^2+\delta^2)^{3/2}} dr = \frac{Q\delta}{2\epsilon_0}\int_0^R\frac{r}{(r^2+\delta^2)^{3/2}} dr\\ = -\frac{Q\delta}{2\epsilon_0} \left.\frac{1}{\sqrt{r^2 + \delta^2}}\right|_{r=0}^R = -\frac{Q\delta}{2\epsilon_0} \left(\frac{1}{\sqrt{R^2 + \delta^2}} - \frac{1}{\delta}\right) = \frac{Q}{2\epsilon_0} - \frac{Q\delta}{2\epsilon_0\sqrt{R^2+\delta^2}}$$

Now there are some cases with which we can check if this result makes sense. If the flat surface extends infinitely, i.e. $R\rightarrow\infty$, we should get $\Phi = \frac{Q}{2\epsilon_0}$, because the total flux through a surface surrounding a charge $Q$ is $Q/\epsilon_0$ from Gauss's law. Since half the flux goes off to the top, half the flux goes down and eventually through the surface (the mantle of the cylinder at $R\rightarrow\infty$ has no contribution). And indeed that's the result we get.

Another case is $\delta \rightarrow 0$. For this case, we should also get $\Phi = \frac{Q}{2\epsilon_0}$, because half the flux will go through the upper hemisphere, and half the flux will go through the lower hemisphere. And indeed, in the limit $\delta \rightarrow 0$ the second term in the result disappears again and we get the same result.

The last case we will check is $\delta \gg R$. This should result in an almost constant field of $E\approx\frac{Q}{4\pi\epsilon_0\delta^2}$ across the whole surface, so the flux should be $\Phi \approx \frac{Q R^2\pi}{4\pi\epsilon_0\delta^2} = \frac{Q R^2}{4\epsilon_0\delta^2}$. We can re-write the second term in the result as a series in $R/\delta$ $$ \frac{Q\delta}{2\epsilon_0\sqrt{R^2+\delta^2}} = \frac{Q}{2\epsilon_0\sqrt{\left(\frac{R}{\delta}\right)^2+1}} = \frac{Q}{2\epsilon_0}\left(1-\frac{(R/\delta)^2}{2} + \mathcal{O}\left(\frac{R}{\delta}\right)^4\right)$$ With that, the flux is $$ \Phi \approx \frac{Q}{2\epsilon_0} - \frac{Q}{2\epsilon_0} + \frac{QR^2}{4\epsilon_0\delta^2} = \frac{QR^2}{4\epsilon_0\delta^2}$$ again in agreement with our expectations.

One more note on the flux through the flat and the curved surface. If we look at the geometry of the problem, for $\delta \gt 0$, all the flux from the charge must enter the semisphere through the flat surface, and exit it through the curved surface (simply because electric field lines of an isolated point charge don't bend). From Gauss's law we know that the total flux through the surface of the semisphere must be 0, as there is no charge inside it. Therefore, the flux through the flat surface and the curved one must be equal in magnitude.

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  • $\begingroup$ First , I know that the electric flux through the flat surface is $-Q/2ε$ and the curved $Q/2ε$ since $δ = 0$ , But If $δ≠0 $ , I think it's the same because it's independent on distance (δ) , But because I didn't study Calculus and Maxwell's equations yet , I really don know how to prove it ! $\endgroup$ – Mohammad Alshareef Apr 12 at 10:32
  • $\begingroup$ How do you know these things if $\delta = 0$? They don't seem right. The point of the limit $\delta \rightarrow 0$ is that the charge is not on the edge of the semisphere, which would not make it as straightforward as for $\delta \neq 0$. $\endgroup$ – noah Apr 12 at 10:53
  • $\begingroup$ Also, have a look at Gauss's law and think about the flux through a complete sphere. $\endgroup$ – noah Apr 12 at 10:55
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    $\begingroup$ You can use Gauss's law for the complete sphere though. $\endgroup$ – noah Apr 12 at 11:12
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    $\begingroup$ @AaronStevens Hah yeah it's probably easier to just use the right triangle of the components of $\vec{E}$ for that, but it had skipped my mind. Maybe I'll correct it later. $\endgroup$ – noah Apr 12 at 15:18

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