Consider the following case:

There is a short electric dipole placed arbitrarily inside a spherical cavity inside a solid,uncharged conducting sphere

We need to find electric field at a point outside this sphere.

I simply tried applying Gauss' law for the point since the charge enclosed is $0$.

$$\oint \vec E\cdot d\vec S=0$$This gives the value of $E$ as $0$. But it could also be so that the surface integral of the the field is 0, but the field at each point is non uniform and non-zero.

Is it possible to find the value of the electric field at any outside point in such case? If yes then how can we do that? Also, can we find the potential due to the dipole at that point, considering it to be $0$ at infinity?

(Note that I am a high school student and have currently not studied higher theorems like uniqueness theorem, Poisson and Laplace etc.)

  • $\begingroup$ Does superposition principle help? $\endgroup$ Apr 24 '19 at 7:14
  • $\begingroup$ Not really. We can't say anything about the charge distribution on the sphere as it the dipole is arbitrarily placed. $\endgroup$
    – user226375
    Apr 24 '19 at 7:38
  • $\begingroup$ I think superposition would help. $\endgroup$
    – my2cts
    Apr 24 '19 at 8:05
  • $\begingroup$ @my2cts can you elaborate? $\endgroup$
    – user226375
    Apr 24 '19 at 16:30
  • $\begingroup$ A dipole field is the sum of the fields of a negative and a positive charge. $\endgroup$
    – my2cts
    Apr 24 '19 at 18:39
  1. In all points of your spherical conductor potential is the same, outer surface included.

  2. External potential is only determined by boundary conditions: $V=\rm const.$ on the outer surface, $V=0$ at infinity.

  3. You can see that outside field has spherical symmetry. Then its flux being zero implies field is zero everywhere (outside conductor).

  • $\begingroup$ Does flux =0 always imply the field is zero ? I don't think so. It could be the case that field is perpendicular to the area vector. $\endgroup$
    – Goarkz
    Jul 2 at 9:48

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