- #1

roam

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## Homework Statement

A long cylindrical wire of radius R

_{0}lies in the z-axis and carries a current density given by:

##j(r)= j_0 \left( \frac{r}{R_0} \right)^2 \ \hat{z} \ for \ r< R_0##

##j(r) = 0 \ elsewhere##

Use the

*differential*form of Ampere's law to calculate the magnetic field B inside and outside the wire.

## Homework Equations

Differential form of Ampere's law: ##\nabla \times B = \mu_0 J##

Curl in cylindrical coordinates:

##\nabla \times B = [\frac{1}{r}\frac{\partial B_z}{\partial \phi}] \hat{r} + [\frac{\partial B_r}{\partial z} - \frac{\partial B_z}{\partial r}] \hat{\phi} + \frac{1}{r} [\frac{\partial}{\partial r} (r B_\phi)-\frac{\partial B_r}{\partial \phi}]##

## The Attempt at a Solution

Could anyone please explain, in the equation above for curl in cylindrical coordinates, which derivative can be non-zero in this case?

If I take this to be the one involving differentiating φ-component with respect to r, then the answer I get for B

_{in}seems to be correct, but B

_{out}is wrong:

##\frac{1}{r} \frac{\partial}{\partial r} (r B_{\phi}) = \mu_0 j_0 (\frac{r}{R_o})^2 \implies B_{in}= \frac{\mu_0 j_0 r^3}{4 R_0^2}##

##\frac{1}{r} \frac{\partial}{\partial r} (r B_{\phi}) = \mu_0 (0) \implies B_{out} = \frac{C}{r}##

##\frac{1}{r} \frac{\partial}{\partial r} (r B_{\phi}) = \mu_0 (0) \implies B_{out} = \frac{C}{r}##

What should I do here?

**P.S.**I am checking my answers by comparing them to the ones I've obtained using the

*integral*form of Ampere's law:

##I_{enc, in} = \int^r_0 \frac{j_0 r^2}{R_0^2} . 2 \pi r dr =\frac{j_0 2 \pi r^4}{4 R_0^2}, \ I_{enc, out} = \frac{j_0 2 \pi R_0^2}{4}##

##\therefore \oint B_{in} .da =B_{in} 2\pi r= \frac{j_0 2 \pi r^4}{4 R_0^2} \implies B_{in}=\frac{\mu_0 j_0 r^3}{4 R_0^2}, \ B_{out}=\frac{\mu_0 j_0 R_0^2}{4 r}##

Any help would be appreciated.