Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A long straight coaxial cable has a solid inner conductor of radius and a thin outer metal tube of radius , with insulation between them, as in the figure. A steady current of flows along the inner conductor and returns along the outer tube. The current is not spread evenly over the inner conductor. Its density is zero on the axis and grows in step with the distance from the axis: . This is a simple model of a current that crowds towards the surface of a wire. The return current is spread evenly round the tube. All the materials are non-magnetic. Take .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. The cable is symmetric, so Ampere's law with circles round the axis gives the field everywhere. The energy needs an integral over , and the push on the tube needs the field that the tube's own current feels.
Step 1. (a). , so .
Step 2. Inside (b, c). The current inside radius is , so . At only one eighth of the current is inside: . At , .
Step 3. Between the conductors (d). at . Outside the tube the enclosed current is zero, and so is the field.
Step 4. Energy (e). Inside, . Between, . Together:
Step 5. Push on the tube (f). Just inside the tube . The tube carries per metre of its width, against the inner current, so it is pushed outwards. Its current feels , so the force per square metre is
Answer: (a) . (b) . (c) . (d) . (e) . (f) .
Watch out. With an even density, a quarter of the current would be inside , and the field there would be ; here the outer layers carry more. And using the full field on the tube doubles the push to .
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