# What is the electric field between two coaxial conducting cylinders when a charged sphere is placed within the inner cylinder?

A long cylindrical capacitor of length $$l$$ consists of an outer conductor of radius $$a$$ and an inner conductor of radius $$b$$, where $$l>>a$$. The outer conductor is earthed, and the inner conductor is hollow, insulated and uncharged. The gaps between both conductors are air filled. A sphere of radius R is charged to a potential $$V$$ far from any other bodies and is then inserted inside the inner conductor of the cylindrical capacitor without touching it.

Draw a diagram showing the field lines and calculate the electric field strength between the two cylinders in terms of the potential $$V$$ and radius $$R$$ of the sphere.

If the sphere has a charge $$+Q$$ distributed on its surface then a charge of $$-Q$$ will be induced on the inner cylinder, and because the cylinder is long we can assume that all the induced positive charge is pushed off to infinity and can be ignored.

That gives us the field inside the inner cylinder, but what about the field between the two cylinders? I'm not sure what those field lines will look like, but my attempt is in the diagram below. Assuming this is correct then I also don't know how to go about calculating $$\vec{E}$$ between the two cylinders. Gauss's law would be easy to apply were the field uniform, but in this case it isn't and I don't know quantitatively how the charge is distributed on the inner cylinder.

Any help is appreciated.

• The cylinders are supposed to be parallel to each other. You've drawn them perpendicular. – lsusr Jun 10 at 23:00
• They're parallel. Note the red radii arrows – Pancake_Senpai Jun 11 at 9:06
• My mistake. I misread the question. – lsusr Jun 11 at 17:53