Problem 3: Current Densities (25 points) (a) A solid cylindrical straight wire o

Question

Problem 3: Current Densities (25 points) (a) A solid cylindrical straight wire of radius a has a current I flowing down it. If that current is uniformly distributed over the outer surface of the wire (none is flowing through the ‘volume of the wire; it's all surface charge), what is the surface current density K? (b) Suppose that current does flow throughout the volume of the wire, in such a way that the volume current density J grows quadratically with distance from the central axis, what then is the formula for J everywhere in the wire? (c) A CD has been rubbed so that it has a fixed, constant, uniform surface electric charge density o everywhere on its top surface. It is spinning at angular velocity w about its center (which is at the origin). What is the surface current density K at a distance r from the center? (d) A sphere (radius R, total charge Q uniformly distributed throughout the volume) is spinning at angular velocity w about its center (which is at the origin.) What is the volume current density J at any point (r, 0, 0) in the sphere? (e) A very thin plastic ring has a constant linear charge density, and total charge Q. The ring has radius R and it is spinning at angular velocity w about an axis through its center (which is at the origin) and perpendicular to the plane of the ring. What is the current I, in terms of given quantities?

Explanation / Answer

Solution (a) : When charge flows over a surface, it is described in terms of Surface Current Density,K.

Mathematically,  K is defined as current per unit width-perpendicular-to-flow. In the question, the width-perpendicular-to-flow will be the circumference of the wire, ie 2?a. Imagine the cylindrical wire to unfurl itself into itself and spread into a sheet, whose width will be the circumference.

Therefore, K=I/2?a.

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