Electric Potential Energy — Question 4

Question 4

Changing the energy reference changes neither force nor accessibility

A charged particle has electric potential energy $U(x)=A x^2+C$, where $A>0$. It starts from rest at $x=b>0$ and moves without other forces. C is an arbitrary reference constant.

1. Find the force and the kinetic energy at a general accessible x.

2. Shift the reference by $C\to C+\Delta C$. Find the turning points and maximum speed for mass m, and demonstrate reference independence.

Question 4 - Solution

Changing the energy reference changes neither force nor accessibility

(a) Method and calculation

$$F_x=-\frac{dU}{dx}=-2Ax,\quad\mathcal E=Ab^2+C.$$$$K(x)=\mathcal E-U(x)=A(b^2-x^2).$$

(b) Interpretation and further calculation

$$K\ge0\Rightarrow|x|\le b,\quad x_{\mathrm{turn}}=\pm b,\quad v_{\max}=\sqrt{2Ab^2/m}.$$$$\mathcal E\prime=\mathcal E+\Delta C,\quad U\prime=U+\Delta C\Rightarrow K\prime=K.$$

Physical check

An absolute negative or positive value of U has no dynamical meaning without a chosen reference; differences and derivatives matter.