Question

In the $\text{xy - plane}$ , the region $y>0$ has a uniform magnetic field $B_1\hat{k}$ and the region $y<0$ has another uniform magnetic field $B_2\hat{k}$ . A positively charged particle is projected from the origin along the positive $\text{y}-\text{axis}$ with speed $v_0=\pi m{s}^{-1}$ at $t=0$ , as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the $\text{x}-\text{axis}$ from below for the first time. If $B_2=4B_1$ , the average speed of the particle, in $m{s}^{-1}$ , along the $\text{x}-\text{axis}$ in the time interval $T$ is ________.

Answer 1

Answer: 2

$(i)$ Average speed along $\text{x}-\text{axis}$

$⟨v_x⟩=\frac{\int |\overrightarrow{v_x}|dt}{\int dt}=\frac{d_1+d_2}{t_1+t_2}$
$(ii)$ We have,
$r_1=\frac{mv}{q{B}_1},r_2=\frac{mv}{q{B}_2}$
Since, $B_1=\frac{B_2}{4}$
$\therefore r_1=4r_2$
Time in $B_1\Rightarrow \frac{\pi m}{q{B}_1}=t_1$
Time in $B_2\Rightarrow \frac{\pi m}{q{B}_2}=t_2$
Total distance along $x-axis$ $d_1+d_2=2r_1+2r_2=2\left(r_1+r_2\right)=2\left(5r_2\right)=10r_2$
Total time $T=t_1+t_2=5t_2$
$\therefore$ Average speed $=\frac{10r_2}{5t_2}=\frac{10mv}{5q{B}_2}\times \frac{q{B}_2}{\pi m}=\frac{10}{5}=2$