\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)
Chapter 15: Electric Potential
2.

Potential Energy of Point Charges

The potential energy of two point charges a distance d apart is
$$PE=k{q_Aq_B\over d}$$

where k=$9\ten9\u{Nm^22/C^2}$ as before. This looks very similar to Coulomb’s Law, but notice the lack of absolute value signs: the potential energy can be positive or negative, and negative PE is considered to be lower than positive PE.

A $+2\u{µC}$ charge starts off a distance 0.4m away from a $+4\u{µC}$ charge, and moves away until it is 1m away. To calculate the change in its potential energy we need the initial and final values: $$\begin{align} \Delta PE &= PE_f-PE_i\\ &= k{q_Aq_B\over d_f} − k{q_Aq_B\over d_i}\\ &= {\color{red}kq_Aq_B}{\color{blue}\left(\frac1d_{\!f} - \frac1d_{\!i}\right)}\\ &= {\color{red}\left(9\ten9\u{Nm^2\over C^2}\right)} {\color{red}(+2\u{\mu C})(+4\u{\mu C})}\\ & \qquad{\color{blue}{\times\left(\frac1{1\u{m}}-\frac1{0.4\u{m}}\right)}}\\ &={\color{red} (+0.072)}{\color{blue}(-1.5)} =−0.108\u{J}\\ \end{align}$$ Because this is negative, the potential energy of the charges decreases, and this makes sense because the two positive charges are moving farther apart, which is exactly what they want to do.

If I replaced the +2µC charge to a –2µC charge, then the change in potential energy would be $\Delta PE = (-0.072)(-1.5) = +0.108\u{J}$. The change is positive, which means a negative charge would not spontaneously start moving away from a positive charge.

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This graph shows how the potential energy changes as the particles get farther apart. When both charges are the same sign then $qAqB>0$, and we use the top curve: the energy gets higher and higher the closer the particles are together. Conversely, for opposite charges we use the bottom curve, and the energy gets lower and lower as the particles get closer. In both cases, however, the potential energy approaches zero* as the distance between the charges gets large.

Multiple Charges

If there are more than two charges, the total potential energy is the sum of the energies from every pair of charges. For instance, if there are four charges, then there are six different pairs to consider: $$\begin{align} PE&={\color{red} k{q_1q_2\over d_{12}}} +{\color{blue} k{q_1q_3\over d_{13}}} +{\color{cyan} k{q_1q_4\over d_{14}}} \\ & +{\color{orange} k{q_2q_3\over d_{23}}} +{\color{purple } k{q_2q_4\over d_{24}}} +{\color{green} k{q_3q_4\over d_{34}}}\\ \end{align}$$

This assumes that the potential energy goes to zero as the two charges move very far apart: in practice, however, only changes in potential energy matter, and so we can use the more general formula $PE=k{q_1q_2\over d}+PE_{\infty}$ where $PE_{\infty}$ is any constant we like, and the physical results will be the same. This is related to how, when calculating the gravitational potential energy mgh of an object, we can measure the height $h$ from any place we want, and the physics will be the same. We won’t exploit this fact with potential energy (having the potential energy be zero when charges are really far apart makes a huge amount of sense), but we will see a similar phenomenon with electric potential later, and that will be a lot more useful.