# Magnetic Field of a current distribution

1. Apr 14, 2016

### motu miah

1. The problem statement, all variables and given/known data

A magnetic lens is made by placing four long, thin current carrying conducting sheets of width $2a$ on the sides of a square, as shown in the figure. The currents in the conducting sheets are distributed uniformly over the sheets, and are directed either into or out of the plane of the page, as shown in the figure. You may neglect the effects of ends and corners.

(a) What are the values of $H_{x}(a^{-},y)$ and $H_{y}(a^{-},y)$ along the boundary just to the left of the current sheet at $x=a$? Likewise, what are the values of $H_{x}(x,a^{-})$ and $H_{y}(x,a^{-})$ along the upper boundary just below the current sheet at $y=a$?

(b) What are $H_{x}$ and $H_{y}$ at the origin?

(c) Find $H_{x}$ and $H_{y}$ in the interior region $-a<x<a$ and $-a<y<a$?

(d) Derive the equation which describes the field lines in the interior region.

(e) Suppose that a beam of positively charged particles is injected into the interior region of the lens so that their velocities are initially along the $z$-axis (out of the page). Discuss how the lens can act to focus the beam. Try to be as quantitative as possible.

2. Relevant equations

3. The attempt at a solution

(a) $H_{x}(a^{-},y) = 0$ and $H_{y}(a^{-},y) = \frac{\mu_{0}I}{2\pi a^{-}}$.

$H_{x}(x,a^{-}) = \frac{\mu_{0}I}{2\pi a^{-}}$ and $H_{y}(x,a^{-}) = 0$.