Equipotential Surfaces — Question 5

Question 5

Equal endpoint potential permits speed changes during the intervening trip

Along an ideal constrained path of length L, $V(s)=V_0+B\sin(2\pi s/L)$ with $B>0$. A positive particle q starts at s=0 with kinetic energy $K_0=2qB$. The guide does no work.

1. Find kinetic energy along the path and at s=L.

2. Find the minimum and maximum speeds for mass m and explain why zero net electric work does not mean zero force throughout.

Question 5 - Solution

Equal endpoint potential permits speed changes during the intervening trip

(a) Method and calculation

$$K(s)=K_0-q[V(s)-V_0]=qB[2-\sin(2\pi s/L)].$$$$K(L)=K_0,\quad W_{0\to L}=0.$$

(b) Interpretation and further calculation

$$v_{\min}=\sqrt{2qB/m},\quad v_{\max}=\sqrt{6qB/m}.$$$$F_s=-q\frac{dV}{ds}=-\frac{2\pi qB}{L}\cos(2\pi s/L).$$

Physical check

Equal endpoint voltages fix net work; only motion entirely on one equipotential makes the electric work vanish on every subsegment.