A coaxial cable takes current to whatever it is connected to, and returns curren

Question

A coaxial cable takes current to whatever it is connected to, and returns current as well (so that the circuit is complete). Here we study such a cable. At the very center is a solid conductor (radius  a ). Outside of this solid conductor is a hollow insulating cylinder, which prevents the charge from jumping from the inner conductor to the outer conductor. On the outside is another conductor- this time a hollow cylinder, with inner radius b  and outer radius c . An opposite current flows in this conductor.

Part A

If the current I  is uniformly distributed in the inner conducting cylinder, what is the current density for r<a ?

Part B

Derive an expression for the magnitude of the magnetic field at points inside the central, solid conductor: r<a .

Express your answer in terms of the variables I, r, and appropriate constants.

Part D

What is the current density in the outer conductor, for b<r<c ? Call the positive direction "out".

Part E

What is the magnitude of the magnetic field for b<r<c ?

A coaxial cable takes current to whatever it is connected to, and returns current as well (so that the circuit is complete). Here we study such a cable. At the very center is a solid conductor (radius  a ). Outside of this solid conductor is a hollow insulating cylinder, which prevents the charge from jumping from the inner conductor to the outer conductor. On the outside is another conductor- this time a hollow cylinder, with inner radius b  and outer radius c . An opposite current flows in this conductor.

(Figure 1)

Part A

If the current I  is uniformly distributed in the inner conducting cylinder, what is the current density for r<a ?

J =

Part B

Derive an expression for the magnitude of the magnetic field at points inside the central, solid conductor: r<a .

Express your answer in terms of the variables I, r, and appropriate constants.

B =

Part D

What is the current density in the outer conductor, for b<r<c ? Call the positive direction "out".

J =

Part E

What is the magnitude of the magnetic field for b<r<c ?

B =

Explanation / Answer

A. For uniform current density, J = I/A = I/pi*a^2
B. From ampere's law, B*2*pi*r = mu*J*pi*r^2 = mu*I*pi*r^2/pi*a^2
B = mu*I*r/2*pi*a^2
D. J' = I/pi(c^2 - b^2)
E. B = mu*[J'*pi(r^2 - b^2) - I]/2*pi*r = mu*I[(r^2 - b^2)/(c^2 - b^2) - 1]/2*pi*r = mu*I(r^2 - c^2)/2*pi*r*(c^2 - b^2)

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