Question

Positive charge Q is uniformly distributed throughout the volume of a dielectric sphere of radius R. A point mass having charge +q and mass m is fired towards the centre of the sphere with velocity v from a point A at distance r(r> R) from the centre of the sphere. Find the minimum velocity v so that it can penetrate R/2 distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.

• A. ${\left[\frac{1}{2\pi {\epsilon }_0}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{4})\right]}^{1/2}$
• B. ${\left[\frac{1}{2\pi {\epsilon }_0}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{8})\right]}^{1/2}$
• C. ${\left[\frac{1}{2\pi {\epsilon }_0}\frac{Qq}{Rm}(\frac{r-R}{r}-\frac{3}{8})\right]}^{1/2}$
• D. ${\left[\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{4})\right]}^{1/2}$

Answer 1

The correct option is B ${\left[\frac{1}{2\pi {\epsilon }_0}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{8})\right]}^{1/2}$

Initial energy are kinetic and potential energy given by-

$K_i=\frac{1}{2}m{v}^2$ and $U_i=\frac{Qq}{4\pi {ϵ}_{o}r}$

Finally at last point , its velocity get reduced to $0$, and potential at a point inside sphere is given by-

$V=\frac{Q}{4\pi {ϵ}_o}\frac{3R^2-r^2}{2R^3}$

At $r=\frac{R}{2}$, $U_f=qV$

$⟹U_f=\frac{Qq}{4\pi {ϵ}_o}\frac{3R^2-\frac{R^2}{4}}{2R^3}$

$⟹U_f=\frac{11}{8}\frac{Qq}{4\pi {ϵ}_{o}R}$

Now applying conservation of mechanical energy-

$K_i+U_i=K_f+U_f$

$⟹\frac{1}{2}m{v}^2+\frac{Qq}{4\pi {ϵ}_{o}r}=0+\frac{11}{8}\frac{Qq}{4\pi {ϵ}_{o}R}$

$⟹m{v}^2=\frac{Qq}{2\pi {ϵ}_o}(\frac{11}{8R}-\frac{1}{r})$

$⟹v^2=\frac{Qq}{2\pi {ϵ}_{o}Rm}(\frac{11}{8}-\frac{R}{r})$

$⟹v^2=\frac{Qq}{2\pi {ϵ}_{o}Rm}(1+\frac{3}{8}-\frac{R}{r})$

$⟹v^2=\frac{1}{2\pi {ϵ}_o}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{8})$

$⟹v=\sqrt{\frac{1}{2\pi {ϵ}_o}\frac{Qq}{Rm}(\frac{r-R}{r}+\frac{3}{8})}$

Answer-(B)