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physicsjones
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Homework Statement
A cylindrical shell of radius a and length 2L is aligned around the z-axis from z= -L ot z = +L. A current I is distributed uniformly on the cylinder and moves around the cylinder's z-axis. Find the magnitude of the magnetic field at the origin.
Homework Equations
Biot-Savart Law [itex] \vec{B} = \frac{\mu_0}{4 \pi} \int\frac{d\vec{I} \times \vec{r}}{r^2}[/itex]
Ampere's Law [itex] \int \vec{B}\dot{} d\vec{l} = \mu_0 I_{enc}[/itex]
The Attempt at a Solution
I've tried this several different ways. First, I tried with ampere's law, the way you would with an infinite solenoid, but that doesn't work since the field isn't perpendicular/constant at the end pieces.
I then tried to use Biot-Savart, but may have done so incorrectly. I got [itex] B = \frac{\mu_0 I}{4 \pi} 2 \pi \int\limits_{-L}^{L} \frac{a}{\sqrt{a^2+z^2}} dz [/itex], which simplifies to (after some Mathematica) [itex] \mu_0 I a \mathrm{arcsinh}(\frac{L}{x}) [/itex], which isn't able to be evaluated at the origin. So I'm stuck.
I've looked at answers like the one given at http://www.phys.uri.edu/~gerhard/PHY204/tsl215.pdf, but have difficulty following and translating them to my problem.