Magnetism Exercise (28-08-2020)

Nice problem. The infinitesimal current element is:

$\vec{dI} = (0,0,j r \, dr \, d\theta)$

The scalar magnetic flux density at radius $r$ is (by convention, I consider $R$ to be the total radius):.

$B = \frac{\mu_0 I_{enclosed}}{2 \pi r} = \frac{\mu_0}{2 \pi r} j \pi r^2 = \frac{\mu_0 r j }{2}$

The vector magnetic flux density is:

$\vec{B} = (- \frac{\mu_0 r j }{2} \sin \theta, \frac{\mu_0 r j }{2} \cos \theta, 0 )$

The infinitesimal flux contribution per unit length is:

$\vec{d F} = \vec{d I } \times \vec{B} = ( -\frac{\mu_0 r^2 j^2 }{2} \cos \theta \, dr \, d \theta, \frac{\mu_0 r^2 j^2 }{2} \sin \theta \, dr \, d \theta, 0)$

Only the $y$ force evaluates to a non-zero net value value over the range $(0 \leq \theta \leq \pi)$ . Evaluating the $y$ force over the half circle $(y \geq 0)$ results in:

$F_y = \int_0^\pi \int_0^R \frac{\mu_0 r^2 j^2 }{2} \sin \theta \, dr \, d \theta = \frac{1}{3} \mu_0 j^2 R^3$