Question

The region between $x=0$ and $x=L$ is filled with uniform, steady magnetic field $B_0\hat{k}$. A particle of mass m, positive charge q and velocity $v_0\hat{i}$ travels along x-axis and enters the region of magnetic field. Neglect gravity throughout the question.
If the value of L if the particle emerges from the region of magnetic field with its final velocity at an angle ${30}^o$ to the initial velocity is $L=\frac{m{v}_0}{Xq{B}_0}$. find X?

Answer 1

As the initial velocity of the particle is perpendicular to the field, the particle will move along the arc of a circle as shown (fig). If r is the radius of the circle, then
$\frac{m{v}_0^2}{r}=q{v}_0{B}_0\Rightarrow r=\frac{m{v}_0}{q{B}_0}$
Also, from geometry, $L=rsin{30}^o\Rightarrow r=2L$
or $L=\frac{r}{2}=\frac{m{v}_0}{2q{B}_0}$