The figure below shows a cylindrical conducting shell carrying current I to the

Question

The figure below shows a cylindrical conducting shell carrying current I to the right. The nonuniform current density in the conductor is given by J = c(r ? R1) for the region R1 < r < R2, where c is a constant. What is the magnitude of the magnetic field at a radial distance rA within the conductor? (Hint: Find the current through an Ampèrian loop of radius r using I(thru) = integral J · dA. Use any variable or symbol stated above along with the following as necessary: ?0. Do not substitute numerical values; use variables only.)

B=?

Explanation / Answer

At radial distance Ra consider an ampere loop
Flux through loop = B*2piRa = mu*I (ampere's law)
B*2piRa = mu*integral(JdA) = mu*integral(c(r - R1))*2pirdr = 2c*pi*mu*integral[r^2 - R1r]dr = 2c*pi*mu[r^3/3 - R1r^2/2]
apply limits, from R1 to Ra
B*Ra = c*mu[[Ra^3/3 - R1*Ra^2/2]-[R1^3/3 - R1^3/2]] = c*mu[Ra^3/3 - R1*Ra^2/2+ R1^3/6]
B = c*mu[Ra^2/3 - R1Ra/2 + R1^3/6Ra]

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