Electrostatics and Magnetostatics
Coaxial cable
Take 10 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.
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Let's consider an infinite cylindrical coaxial cable of radii .
The total intensity goes in one way through the interior conductor and comes back in the other way through the exterior conductor.
Let us suppose the current is evenly distributed in the volume of the interior conductor, and distributed on the surface of the exterior conductor.
Question
Compute the magnetic field in any point of space.
Hint
Think about Ampère's circuital law. First do a study of the symmetry.
Solution
Symmetry and invariance studies give :
.
Let's apply Ampere's circuital law to a circle of radius that interlaces the wire.
Let's consider the cases :
: (No current interlaced)

If :
Hence :

If :

If : (interlaced current globally equal to zero)
Question
Verify the interface conditions.
Hint
Solution
We notice that the field is continuous in and in (property of a volume distribution of currents).
In : the magnetic field is discontinuous, and :
The surface density on the cylinder of radius is expressed by :
The interface conditions are verified during the crossing of a surface travelled by a surface current :