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# Electrostatics: Charge density in a coaxial cable.

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We have a long coaxial cable with an inner solid wire of radius a and outer
metal shell of radius b. On the inner wire, the volume charge density is given by ks2 .
On the outer shell, the linear charge density (along the axis) is given by &#955;.
(a) Draw a picture that illustrates this arrangement.

(b) What is the linear charge density &#955; that will give E = 0 outside the cable?

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electrostatics
We have a long coaxial cable with an inner solid wire of radius a and outer metal shell of radius b. On the inner wire, the volume charge density is given by ks2 . On the outer shell, the linear charge density (along the axis) is given by λ.
(a) Draw a picture that illustrates this arrangement.

(b) What is the linear charge density λ that will give E = 0 outside the cable?
Consider a length L of the cable.
The volume charge density on the inner solid cylindrical wire is  = ks2.
Consider a cylindrical sell of radius s and infinitesimally small thickness ds. The volume of such a shell of length L is given by

And thus the charge on this shell will be

Or
Thus the total charge on the length L of the inner wire is given by integrating this elementary charge for radius o to a of the wire and hence

Or
And hence the linear charge density on the inner wire along the length is given by
------------------------ (1)
Now as the magnitude of the electric field strength is given by the flux per unit area normal to the field, for the field outside of the cable to be zero, the flux outside of the outer shell must be zero. For this to be happen the total flux from the charge on the inner wire must be terminated to the charge on the outer shell and for that, as the system is cylindrically symmetric, the linear charge density on the outer shell must be equal to that on the inner wire in magnitude and opposite in sign. Thus the linear charge density on the outer shell is given by

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!