Electric Potential Energy — Question 10

Question 10

A time-dependent prescribed field requires an explicit energy-source term

A one-dimensional prescribed field is $E(t)$, spatially uniform. Assign a charge q the energy $U(x,t)=-qE(t)x$. Assume the electric force dominates and use $F_x=-\partial U/\partial x$.

1. Derive the rate of change of K+U for a freely moving particle.

2. Now hold the particle at fixed $x=x_0$ while $E(t)=E_i+\alpha t$ changes for duration T. Find its assigned energy change and electric mechanical work.

Question 10 - Solution

A time-dependent prescribed field requires an explicit energy-source term

(a) Method and calculation

$$\frac{dK}{dt}=qE(t)\dot x,\quad\frac{dU}{dt}=-qE(t)\dot x-q\dot E(t)x.$$$$\frac{d(K+U)}{dt}=-q\dot E(t)x=\frac{\partial U}{\partial t}.$$

(b) Interpretation and further calculation

$$x=x_0:\quad\Delta U=-q\alpha Tx_0.$$$$W_e=\int qE(t)\,dx=0.$$The assigned particle-plus-interaction energy changes because the external source varies the field; time independence was essential to the earlier conservation rule.

Physical check

The energy zero is fixed by the supplied U; adding an arbitrary function of time would change its bookkeeping rate but not the force.