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I am very familiar with using Ampere's Law to find the B field inside a solenoid ($\mu_0nI(t)$). Then I can use Faradays to find the E field inside ($\propto\dot I$).

I want to get this result this way using Maxwell's equations inside the solenoid for a harmonic current ($\Re[I_0e^{-i\omega t}]$). Maxwells equations will be $$\nabla\cdot E=\nabla\cdot B=0$$ $$\nabla\times E=i\omega B$$ $$\nabla \times B=-i\omega\mu\epsilon E$$ Which will reduce to $(\nabla^2+k^2)\begin{bmatrix}E\\B\end{bmatrix}=0$. From symmetry, I know that B is only in z and E is only in $\theta$ (cylindrical coords). When I solve the equation for B, since $\nabla\cdot B=0\rightarrow dB/dz=0\rightarrow B(z)=\text{Constant}$

But when I solve helmholtz equation in cylindrical coordinates, I get that $$B=AJ_0(k\rho)$$ SO the field depends radially on location from the center. This does not agree with the constant $\mu nI$. Is this approach correct? If I impose the boundary conditions, (which I am unsure of), will I get something that is not a bessel function?

EDIT: "VEry long" solenoid so edge effects are neglected

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  • $\begingroup$ Could you give your reasoning for why the $\mathbf{B}$-field is only in the $\hat{z}$-direction and why the $\mathbf{E}$-field is only in the $\hat{\theta}$-direction? $\endgroup$
    – Arturo don Juan
    Commented Apr 24, 2016 at 17:30
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    $\begingroup$ Symmetry. If I have current wrapped around in theta direction, then the field is in z. Then since E is curl B, it will be in theta $\endgroup$
    – yankeefan11
    Commented Apr 24, 2016 at 17:31
  • $\begingroup$ You have found $\frac{d B}{dz} = 0$, this gives $B$ independent of $z$ but it may be a function of $\rho$? $\endgroup$
    – jim
    Commented Apr 24, 2016 at 18:41
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    $\begingroup$ You may be interested in the following reference: Electrically induced magnetic fields; a consistent approach Brian Batell and Andrew Ferstl Am. J. Phys. 71, 925 (2003) $\endgroup$
    – jim
    Commented Apr 24, 2016 at 19:30

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