I´ve been solving the following problem and I need some guidance. The problem asks to calculate all the components of the field momentum of a circular toroidal coil of mean radius a and N turns with a small cross section area A (which is small compared to a). The toroid has corrent I flowing in it and there is a point charge Q located at its center.
Atempted Solution
The electromagnetic momentum field is given by
$$P_{field}=\epsilon_{0}\int_{V} \textbf{E}\times \textbf{B}$$
Calculating the electric field of the point charge we have $$\textbf{E}=\frac{Q}{4\pi\epsilon_0 r^2}\hat{r}$$
and given that $a>>A$ i wrote the magnetic field of the toroid as
$$\textbf{B}=\frac{\mu_0 NI}{2\pi\epsilon_0 a}\frac{\sin{\theta}}{a}\delta{(\cos{\theta})}\delta{(r-a)}\hat{\phi}$$
performing the integral the result is $$\epsilon_{0}\int_V \textbf{E}\times \textbf{B}=\epsilon_{0}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{a} (\frac{Q}{4\pi\epsilon_0 r^2}\hat{r})\times \left(\frac{\mu_0 NI}{2\pi\epsilon_0 a}\frac{\sin{\theta}}{a}\delta{(\cos{\theta})}\delta{(r-a)}\hat{\phi}\right)d^3x=\frac{\mu_0 QNI}{4\pi a^2}\hat{z}$$
however this is incorrect as the expected answer is $$\frac{\mu_0 QNIA}{4\pi a^2}\hat{z}$$ I know that my mistake comes from treating the toroid as a small ring with the current I localized in the mean radius a, thus the area A not appearing after computing the integral. I´ve been told that by taking some aproximations, such as directly asumming the momentum only has a $\hat{z}$ component, this integral can be performed more easily, but i don´t know why i can assume that directly (other than plotting the momentum vector field maybe?). A way to correct my result that I can think of is to establishing the quantity $\frac{\mu_0 QNI}{4\pi a^2}\hat{z}$ as the momentum field of a ring with radius a and the integrating over the area A of the toroid. Something like: $$\int_A P_{ring}=\int \frac{\mu_0 QNI}{4\pi a^2}\hat{z} dA$$ Would this be conceptually correct? Any help would be greatly appreciated!! Thanks.
