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Bhumble
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Homework Statement
A point charge q is a distance a > R from the axis of an infinite solenoid (radius R, n turns per unit length, current I). Find the linear momentum and the angular momentum in the fields. (Put q on the x axis, with the solenoid along z; treat the solenoid as a nonconductor, so that you don’t need to worry about the induced charges on its surface).
[Answer: [tex] \vec{p_{em}} = \frac{\mu_0 q n I R^2}{2a} and \vec{L_{em}} = 0 [/tex]
Homework Equations
[tex] \vec{\rho_{em}} = \epsilon_0 \vec{E}\times\vec{B} [/tex]
The Attempt at a Solution
So I know that I'm suppose to get the linear momentum density and then integrate over surface area to get the momentum.
And for linear momentum density I just take r X (E x B) then integrate over the surface area.
I have [tex] B(s>R) = 0 and B(s<R) = \mu_0 n I \hat{z}[/tex]
The problem I'm having is setting up the electric field.
I know that it is [tex] \vec{E} = \frac{q}{4 \pi \epsilon_0 r^2} \hat{r}[/tex]
r^2 = (x-a)^2 + y^2 + z^2 and since there is translational invariance along [tex]\hat{z}[/tex] I just dropped it altogether. But for [tex]\hat{r}[/tex] I'm unsure how to define it so that I can get a cross product. I was thinking that it should be equal to [tex] r cos \theta + r sin \theta [/tex] then take A X B = ABcos[tex]\theta[/tex] but since cross product is not distributive I'm just unsure.
I'm very confused about how to approach the geometry of the situation and this is not the first time this has been an issue for me. Any help is appreciated.