Question

Find the magnetic field a distance r from the center of a long wire that has radius a and carries a uniform current per unit area j in the positive z-direction.

PART A

First find the magnetic field, B? out(r? ),outside the wire (i.e., when the distance ris greater than a). (Figure 1)

Express B? out(r? ) in terms of the given parameters, the permeability constant ?0, the variables a, j (the magnitude of j? ), r,?, and z, and the corresponding unit vectorsr^, ?^, and k^. You may not need all these in your answer.

PART B

Part B

Now find the magnetic field B? in(r? )inside the wire (i.e., when the distance r is less than a). (Figure 2)

Express B? in(r? ) in terms of the given parameters, the permeability constant ?0, the distance r from the center of the wire, and the unit vectorsr^, ?^, and k^. You may not need all these in your answer.

Answer 1

a) Using ampere law:

\(\int B d l=\mu_{0} I_{n e t}\)

so:

\(B(2 \pi r)=\mu_{0}(j A)=\mu_{0} j\left(\pi R^{2}\right)\)

\(B=\frac{\mu_{0} j R^{2}}{2 r}\)

the direction is in \(\hat{\theta}\)

\(\vec{B}=\frac{\mu_{0} j R^{2}}{2 r} \hat{\theta}\)

b)

Using ampere law:

\(\int B d l=\mu_{0} I_{n e t}\)

so:

\(B(2 \pi r)=\mu_{0}(j A)=\mu_{0} j\left(\pi r^{2}\right)\)

\(B=\frac{\mu_{0} j r}{2}\)

the direction is in \(\hat{\theta}\)

\(\vec{B}=\frac{\mu_{0} j r}{2} \hat{\theta}\)