Electrical potential energy
Read Tipler and Mosca (2004) chapter 23 (Electric
potential); Pelcovits and
Farkas (2024) chapter 10 (Electrostatic) pp.289–302.
There are three topics in this unit: electric potential
energy; electric potential, and conservation of electric energy.
You should be able to describe the electric potential energy of a system.
Electric potential energy between point charges
The electric potential energyEnergy is measured in joules, , where \(\qty{1}{\joule}=\qty{1}{\newton\meter}\).
Named for James Prescott Joule, English physicist, mathematician, and
brewer, who studied energy. See
of a system of two point charges equals the amount of
work required for an external force to bring the point charges to their
current positions from infinitely far away.
Recall the magnitude of the force between two point charges isEquation 1 on the official
AP Physics C Electricity & Magnetism equation sheet
\[\begin{equation}
|F| = \dfrac{k q_1 q_2}{r^2} = \dfrac{1}{4\pi\epsilon_0} \dfrac{q_1
q_2}{r^2}.
\end{equation}\qquad{(1)}\]
The force between the point charges is repulsive (\(+\hat{r}\) direction) if the charges are both of the same sign; or it is attractive (\(-\hat{r}\) direction) if the charges are opposite sign.
Consider two positive charges. At \(r=\infty\), the charges are so far apart neither knows anything about the other. We now wish to know how much work it takes to drag one of the charges from infinity to \(r\).
\[ \text{work} = \text{force} \cdot \text{distance} \] \[ dW = \mathbf{F} \cdot d\mathbf{s} \] \[ W = \int \mathbf{F} \cdot d\mathbf{s} \qquad{(2)}\]
The change in potential energy associated with a conservative force acting on an object as is moves from \(A\) to \(B\) is defined as \[\begin{equation} \Delta U = U_B - U_A = -\int_A^B \mathbf{F} \cdot d\mathbf{s} = -W \end{equation}\qquad{(3)}\]
Substituting eq. 1into eq. 2, and applying eq. 3, gives the
general form for the electric potential energy between two
charged objectsThis is equation 7 on the oficial
AP Physics C Electricity & Magnetism equation sheet
: \[\begin{align}
\Delta U &= -\int_\infty^r \dfrac{k q_1 q_2}{s^2} ds \\
&= k q_1 q_2 \int_\infty^r -\dfrac{1}{s^2} ds \\
&= k q_1 q_2 \left. \dfrac{1}{s} \right|_\infty^r \\
&= k q_1 q_2 \left[ \dfrac{1}{r} - \dfrac{1}{\infty} \right] \\
U_E &= \dfrac{k q_1 q_2}{r}.
\end{align}\]
This can also be written using \(k = \frac{1}{4\pi\epsilon_0}\): \[\begin{equation} U_E = \dfrac{1}{4\pi\epsilon_0}\dfrac{q_1 q_2}{r} = \dfrac{k q_1 q_2}{r}. \end{equation}\]
\(U_E\) here is expected to be
positiveSign check! Does it make sense this should be positive
for two like charges? Yes.
for two charges of the same sign, reflecting the
idea that you must do work, supply energy to force them together when
they would rather be far, far apart.
When two charges of opposite sign are used, \(U\) would be negative, indicating a release of energy as they are attracted to one another.
Superposition applies!
The total electric potential energy of a system can be determined by finding the sum of the electric potential energies of the individual interactions between each pair of charged objects in the system.
For example, in a system of three charges, \(Q_1\), \(Q_2\), and \(Q_3\), you would find the energy for each pair: \[\begin{equation} U = \dfrac{k Q_1 Q_2}{r_{12}} + \dfrac{k Q_2 Q_3}{r_{23}} + \dfrac{k Q_3 Q_1}{r_{31}} \end{equation}\]
See also
- add here