1 The Calculation

1.1 Introduction

We wish to calculate the magnetic field of a long solenoid with N turns per unit length carrying a current which has the same value at every point of the solenoid as in Jackson problem 6.24. We will solve for the case where the current is ReIe ^{- i$\omega$t}, and use this result to calculate the field the of triangular wave current, which is periodic and has the value.

1.2 The Green's function

The solution to the two-dimensional Helmholtz equation is

$$\nabla^{2}_{} \pi \delta \vec{r} \vec{r'}$$ (1)

which has only outgoing waves at infinity is

$$\rho \phi \rho \phi \pi \sum_{m=-\infty}^{\infty} \rho_{<}^{} \rho_{\gt}^{} \phi \phi$$ (2)

1.3 The magnetic Field

For the solenoid the vector potential is then

$$\phi {\frac{i 2 \pi^2 a N I(\omega) }{c}} \rho_{<}^{} \rho_{\gt}^{}$$ (3)

where $\rho_{<}^{}$ is the smaller of a and $\rho$ , and $\rho_{\gt}^{}$ is the larger. Taking curl of $\vec{A}$ , we get

$${\textstyle\frac{1}{\rho}} {\frac{\partial}{\partial \rho}} \rho \rho$$ (4)

We can use the identify for Bessel functions C₁ , that

$${\textstyle\frac{1}{x}} {\frac{d}{dx}}$$ (5)

to give the magnetic field

$${\frac{i 2 \pi^2 k a N I(\omega)}{c}} \left \{ \begin{array} {ll} J_0(k \rho) H_1^{(1)}(ka) & \rho < a \ H_0^{(1)}(k\rho ) J_1(ka) & \rho \gt a \ \end{array}\right.$$ (6)

As a check, if $\omega$ $\rightarrow$ 0, J₁(ka) $\rightarrow$ 0, and the $\rho$ > a component goes to zero, while H₁⁽¹⁾(ka) $\rightarrow$ - 2i/($\pi$ka) , and we get the expected quasistatic result, B_z = 4$\pi$NI($\omega$)/c for $\rho$ < a .

1.4 Triangular Current Results

For a triangular wave current, I have

$${\frac{8 I_0}{\pi^2}} \sum_{n \ \rm{odd}}^{} {\frac{\cos(n \omega_0 t)}{n^2}}$$ I(t) =

$${\frac{8 I_0}{\pi^2}} \sum_{n \ \rm{odd}}^{} {\frac{e^{i n \omega_0 t}}{n^2}}$$ = (7)

where 2$\pi$/$\omega$ is the period of the oscillation. Plugging this in, the magnetic field as a function of time becomes

$$\rho \sum_{n \ {\rm odd}}^{} {\frac{i 16 \omega_0 a N I_0}{n c^2}} \omega_{0} \left \{ \begin{array} {ll} J_0(n \omega_0 \rho/c) H_1^{(1)}(n \om... ...{(1)}( n \omega_0 \rho /c) J_1(\omega_0 a/c) & \rho \gt a \ \end{array}\right.$$ (8)

Defining x = $\rho$/a , $\beta$ = $\omega_{0}^{}$a/c , $\tau$ = $\omega_{0}^{}$t , and B₀ = 4$\pi$NI₀/c this can be written

$$\rho {\frac{B_0 4\beta}{\pi}} \sum_{n \ {\rm odd}}^{} {\textstyle\frac{1}{n}} \left \{ \begin{array} {ll} J_0(n \beta x) [ J_1(n \beta) \sin(n \... ... ) \sin(n \tau) - N_0(n \beta x)\cos(n \tau) ] & \rho < a \ \end{array}\right.$$ (9)