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There is a question in my textbook that says:

A rubber balloon is given a charge $Q$ distributed uniformly over its surface. Is the field inside the balloon zero everywhere if the balloon does not have a spherical surface?

Intuitively i think the answer is no, but i am not able to prove it conclusively. The best i could say is that if the balloon were spherical, and a minor depression was made in its surface somewhere, it would destroy the symmetry and hence create non-zero field at all points.

Is there a better logical proof for it? Not too advanced mathematics, but i can understand integration (for example i understand the Gauss's Law) and am willing to learn a few more concepts, if necessary. A mathematical proof, not very advanced, would be a delight to me.

Thanks!

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Use superposition.

Start with your actual balloon: spherical except on place were it dips inside to be closer to the center.

Next, imagine a uniformly charged perfectly spherical shell with the same charge density.

Then imagine another object (a small object) with a positive charge where the balloon is bent inwards, and a negative charge where the the balloon isn't lined up with the spherical shell.

The charge of both the small object and the spherical shell add up to the charge of the actual balloon. So the field due to each adds up to the field due to the actual balloon.

The field of the spherical shell is zero, so we only need to find the field of the small object.

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