Chapter 1

# 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 : Previous
Ampère's circuital law
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Coaxial cable