Question

A particle of charge 1mC and mass 1gm is projected from origin with velocity 20 m/s towards positive x-axis in a nonuniform magnetic field $\overrightarrow{B}=-b_{0}x\hat{k},$ where $b_0=\frac{1}{10}T/m$ and x is in metre. If the maximum positive x-coordinate of the particle during its motion is $4x_0$ (in cm) then $x_0$ is (Neglect gravity)

Answer 1

Given $q={10}^{-3}C;m={10}^{-3}㎏$
$\overrightarrow{B}=-b_{0}x\hat{k};b_0=\frac{1}{10}T/m$
$v=20㎧$
To find $x_0$
Solution:-
Force acting on a charged particle moving in magnetic field$=q\overrightarrow{v}\times \overrightarrow{B}$
Magnetic force at orgin is in $+y$ direction. The particle moves in $x-y$ plane. The force is always perpendicular to the velocity so it will only change the direction of velocity vector but not hte magnitude. As magnetic field is non uniform its path is not a perfect circle. Let at point $A(x,y,)$ velocity makes angle $\theta$ with x-direction
$F_m=q{v}_{0}B\sin 90=q{v}_0{b}_{0}x\to 1$
Velocity along x direction $v_x=\frac{dx}{dt}=v_0\cos \theta \to 2$
Velocity along y direction$v_y=\frac{dy}{dt}=v\sin \theta$
Now,
$F_y=F_m\cos \theta$
$\Rightarrow a_y=\frac{F_m}{m}\cos \theta$
$\Rightarrow \frac{d{v}_y}{dt}=\frac{b_{0}x}{m}\times q{v}_0\times \cos \theta$
$\Rightarrow \frac{d{v}_y}{dx}\frac{dx}{dt}=\frac{b_{0}x}{m}q\left(v_0\cos \theta \right)$
$\Rightarrow \frac{d{v}_y}{dx}\times v_0\cos \theta =\frac{b_{0}x}{m}q\left(v_0\cos \theta \right)$(from equation $2$)
$\Rightarrow d{v}_y=\left(\frac{b_{0}x}{m}\right)xdx$
At maximum displacement along x direction velocity of particle is along y direction i.e., $v_y=v_0$
$\Rightarrow {\int }_0^{v_0}d{v}_y=\frac{b_{0}q}{m}{\int }_0^{x_{max}}xdx$
$\Rightarrow v_0=\frac{b_{0}q}{m}\frac{x_{max}^2}{2}$
$\Rightarrow x_{max}=\sqrt{\frac{2v_{0}m}{q{b}_0}}$
$\Rightarrow x_{max}=\sqrt{\frac{2\times 20\times {10}^{-3}}{{10}^{-3}\times 1/10}}=\sqrt{{\left(2\right)}^2\times {\left(10\right)}^2}$
$\Rightarrow x_{max}=20m=20\times 100㎝$
$\Rightarrow x_{max}=4x_{0}420\times 100㎝$
$\Rightarrow x_0=5\times 100㎝$
$\Rightarrow x_0=5m$