# Ampere's law on a long wire with varying current density [closed]

On a question from my book:

A long straight wire with a circular cross section of radius $R$ carries a current $I$. Assume the current density is not constant over the cross section of the wire, but rather varies as $J=\alpha r$ where $\alpha$ is a constant. Given $I, R$

Find $\alpha$

Find the magnetic field as a function of r both inside and outside the wire

I think it's just the calculus parts confuses me. My attempt:

$$J=\alpha r' = \frac{dI}{dA}$$ $$dI = 2 \pi r'^2 dr' \alpha$$

$$I = 2 \pi \alpha \int_0^R r'^2 dr'$$ $$I = 2 \pi R^3 \alpha /3$$ $$\alpha = \frac{3I}{2 \pi R^3}$$

from here you just use ampere's and I believe there's no variance issues?

$$\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{in}$$

apply J=I/A

$$B 2 \pi r = \mu_0 \alpha r A$$

$$B = \frac{\mu_0 \alpha A}{2 \pi}$$]

$$B = \frac{3 \mu_0 I r^2}{2 R^3}$$

Is this right? The units seem to line up so I'm hopeful.

Outside the wire is treated as the general uniform wire case I assume and am not too worried about that.

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## closed as off-topic by tpg2114, Dimensio1n0, John Rennie, Qmechanic♦Nov 9 '13 at 16:31

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To check the first part you can check the units and see if they work out. Then you can pretty much only be off by a prefactor. To check the prefactor I would just recalculate $I$ in terms of $\alpha$ and $R$. That is, I would evaulate $\int \alpha r 2 \pi r dr$, plugging in your claimed expression for $\alpha$, and see if this does indeed give me $I$.
To test the value for $B$, you can check dimensions. Your expression has the same dimensions of $\mu_0 I /R$ so that is good. Another thing you can do is test the limits $r \to 0$ and $r \to R$. Do you get what you expect in these cases?