Question

The current density in a cylindrical wire of radius R = 2.0 mm is uniform across a across section of the wire and is J = 2.0 times 10^5 A/m^2. What is the current through the outer portion of the wire between the radial distances R/2 and R and shown below Suppose, instead, that the current density through a cross section varies with the radial distances r as J = ar^2 in which a = 3.0 times 10^11 A/m^2 and r is in meters. What now is the current through the same outer portion of the wire?

Answer 1

Here,

R = 2 mm

J = 2 *10^5 A/m^2

for the current through wire between R/2 and R = J * area

the current through wire between R/2 and R = J * pi (R^2 - (R/2)^2)

the current through wire between R/2 and R = 2 *10^5 * pi * (0.002^2 - (0.002/2)^2)

the current through wire between R/2 and R = 1.885 A

the the current through wire between R/2 and R is 1.885 A

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b)

J = a r^2

J = 3 *10^11 * r^2

For the current

I = integration(J * 2pir * dr)

I = integration(3 *10^11 * 2 * pi * r^3 . dr)

I = 3 *10^11 * 2pi * [r^4/4]

I = 3 *10^11 * 2pi * [0.002^4 - 0.001^4]/4

I = 7.07 A

the current in the loop is 7.07 A