A Point Charge in an Electric Field — Question 10

Question 10

A rotating prescribed field can return velocity to zero while transporting a particle

A particle of mass m and charge $q>0$ starts from rest at the origin in a spatially uniform prescribed field $\mathbf E(t)=E_0(\cos\omega t,\sin\omega t)$, with $\omega>0$. Assume the electric force dominates and motion remains nonrelativistic. Define $a_0=qE_0/m$.

1. Integrate to obtain both velocity and position components.

2. Evaluate them after one period $T=2\pi/\omega$ and check the net work from the kinetic-energy change.

Question 10 - Solution

A rotating prescribed field can return velocity to zero while transporting a particle

(a) Method and calculation

$$v_x=\frac{a_0}{\omega}\sin\omega t,\quad v_y=\frac{a_0}{\omega}(1-\cos\omega t).$$$$x=\frac{a_0}{\omega^2}(1-\cos\omega t),\quad y=\frac{a_0}{\omega^2}(\omega t-\sin\omega t).$$

(b) Interpretation and further calculation

$$\mathbf v(T)=\mathbf0,\quad\mathbf r(T)=\left(0,\frac{2\pi a_0}{\omega^2}\right).$$$$W_{0\to T}=\Delta K=0.$$The velocity vector completes a cycle, while its nonzero time average produces a net displacement.

Physical check

The source driving this time-dependent field is external; the calculation concerns mechanical work on this particle over one complete pulse.