Chapter 1

Electrostatics and Magnetostatics

Uniform rotation of a cylinder evenly charged in volume

Take 20 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Let be a cylinder of revolution around the axis , of radius .

Its length is very big compared to .

is evenly charged with a volume density .

It is set in motion to rotate around axis, with an angular speed .

This rotation does not depend on time, and it does not affect the repartition of charges in .

Question

Find the electric field .

Solution

Let's use Gauss' law : (the electric field is radial)

We choose a cylinder of radius and axis (Oz).

  • For  :

  • For  :

We have verified that the electric field in continuous when crossing the cylinder (In )

Question

Find the magnetic field at any point in space.

Hint

The currents are ortho-radial. Use the same reasoning as for a classical solenoid and use Ampere's circuital law.

Solution

Let's use Ampere's circuital law :

The magnetic field is oriented as the axis of the solenoid, and we know it is equal to zero outside.

Let us choose a rectangular loop of lenght .

Coaxial cable

One of its sides is parallel to the axis and is inside the solenoid, whereas the other one is outside the solenoid.

At a distance from the (Oz) axis, the current density is given by :

Thus, for , Ampere's circuital law gives :

Finally :

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