1. Given potential

   $V(r)=Fr$ where $F>0$ is a constant.

   The force is $\vec F_r=-\frac{dV}{dr}\hat r=-F\hat r$ so its magnitude is constant and directed towards the origin.

2. Condition for circular motion

   For a circular orbit of radius $R$, centripetal force is provided by this force: $\frac{mv^2}{R}=F$ Hence, $mv^2=FR \quad \Rightarrow \quad v^2=\frac{FR}{m}$

3. Bohr quantization condition

   According to Bohr model, $mvr=\frac{nh}{2\pi}$ For radius $R$, $mvR=\frac{nh}{2\pi}$

4. Find dependence of $R$ on $n$

   From step 2, $v=\sqrt{\frac{FR}{m}}$ Substitute into Bohr condition: $mR\sqrt{\frac{FR}{m}}=\frac{nh}{2\pi}$

   Simplify: $\sqrt{mF}\,R^{3/2}=\frac{nh}{2\pi}$

   Therefore, $R^{3/2}\propto n$ so $R\propto n^{2/3}$

   Thus the first part matches option B.

5. Find dependence of $v$ on $n$

   Since $v^2=\frac{FR}{m}$ we have $v\propto R^{1/2}$

   Using $R\propto n^{2/3}$, $v\propto n^{1/3}$

   Hence option B is correct, and option A is incorrect.

6. Find exact expression for radius

   From $\sqrt{mF}\,R^{3/2}=\frac{nh}{2\pi}$ we get $R^3=\frac{n^2h^2}{4\pi^2 mF}$

   Thus, $R=\left(\frac{n^2h^2}{4\pi^2 mF}\right)^{1/3}$

7. Total energy of the orbit

   Total energy is $E=T+V$

   Kinetic energy: $T=\frac12 mv^2$ Using $mv^2=FR$, $T=\frac12 FR$

   Potential energy: $V=FR$

   Therefore, $E=\frac12 FR+FR=\frac32 FR$

   Substitute $R$: $E=\frac32 F\left(\frac{n^2h^2}{4\pi^2 mF}\right)^{1/3}$

   Rearranging, $E=\frac32 \left(\frac{n^2h^2F^2}{4\pi^2 m}\right)^{1/3}$

   This matches option C.

8. Check option D

   Option D has coefficient $2$ instead of $\frac32$, so it is incorrect.

9. Final selection

   Correct options are:

   • B
   • C

10. Comparison with stored answer

Stored correct answer: B, C

This matches exactly.