Electrostatics and Magnetostatics
Uniform rotation of a cylinder evenly charged in volume
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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 .
Hint
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 orthoradial. 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 .
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 :