A copper wire (radius R and length L) carries a current density of magnitude J(r

Question

A copper wire (radius R and length L) carries a current density of magnitude J(r)=ar^2. For each part you must find a final symbolic answer before using the numbers listed below to find a numerical answer.

R= 1.0 mm L= 1.0 m a= 1.5x10^5 A/m^4

(a) What is the current i(shell) contained in the hollow shell of wire defined by R/2<r<R ?

(b) What is the total current i(tot) running through the wire?

(c) Given your answer to (b), what potential difference V must be applied between the ends of the wire?

Hint: For this part, you may have to look up the resistivity of copper.

Explanation / Answer

(a) current i in the shell =J*2r*dr (limits from R/2 to R ) = 2a r^4 / 4 = (*a/2)*[R^4-R^4/16] = 15aR^4/32

= 2.2 * 10^-7 A

(b) total current = J*2r*dr (limits from 0 to R ) = 2a r^4 /4= (*a/2)*R^4= 2.356 *10^-7

(c) we know R=l/A

Resistivity of copper wire = 1.68×108

R=1.68×108 *1/R^2 = 0.0053

Potential difference = I*R = 2.356*10^-7 * 0.0053= 1.259 * 10^-9 V

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