Consider a long, solid, cylindrical wire of radius a = 3.1 mm. The current densi

Question

Consider a long, solid, cylindrical wire of radius a = 3.1 mm. The current density vector J inside this wire points everywhere in the direction of the central axis of the wire. Its magnitude depends on the distance r from the central axis of the wire according to J = J(r) = J0r/a where J0 is a constant. (a) What are the units of J0? (b) Use Ampere's law to find an expression for the magnitude of the magnetic field B(r) inside the wire in terms of J0, a, and r. (c) Check your answer to part (b) by considering the following limits: (i) We expect the field at the center of the wire to he equal to zero. Why? Explain. Is the expression for the magnetic field you derived in part (b) consistent with this prediction? (ii) We expect that B(r) will increase as J0 is increased. Why? Explain. Is the expression for the magnetic field you derived in part (b) consistent with this prediction? (iii) If we keep r fixed but increase the radius a of the wire, we expect the field at r to decrease. Why? Explain. Is the expression for the magnetic field you derived in part (b) consistent with this prediction?

Explanation / Answer

a) Ampere

b)B = ?j/2?r

C)

1) equal and opposite induction acts in side along radius. no

2) they are directly proportional. yes

3) current density reduces, hence the reduction in B, yes

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