Question

A positively charged particle of mass m and charge q is projected on a rough horizontal x-y plane surface with z-axis in the vertically upward direction. Both electric and magnetic fields are acting in the region and given by $\overrightarrow{E}=-E_0\hat{k}$ and $\overrightarrow{B}=-B_0\hat{k}$, respectively. The particle enters into the field at $(a_0,0,0)$ with velocity $\overrightarrow{v}=v_0\hat{j}$. The particle starts moving in some curved path on the plane. If the coefficient of friction between the particle and the plane is $\mu$. If the distance travelled by the particle when it comes to rest is given by $l=\frac{m{v}_0^2}{x\mu (mg+q{E}_0)}$. Find x.

Answer 1

$-m\frac{dv}{dl}\frac{dl}{dt}=\mu N$

$\Rightarrow -mv\frac{dv}{dl}=\mu (mg+q{E}_0)$

$\Rightarrow -m{\int }_{v_0}^{0}vdv=\mu (mg+q{E}_0){\int }_0^{l}dl$

$\Rightarrow l=\frac{m{v}_0^2}{2\mu (mg+q{E}_0)}$