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

Electric Potential

There are a couple ways to describe the way a source charge distorts space. One way is to consider how the electric potential energy of a system would change at any particular location if a hypothetical target charge $q_t$ were placed there. For example, consider a single point source charge $q_s$, and a location (marked by the star) a distance $d$ away. Suppose we want to describe the distortion of space at the star by asking "How would the energy of the system increase if I put a charge there?" We imagine placing a target charge (sometimes called a test charge) at the star. The potential energy of the system then changes from zero (a single charge has no potential energy) to $$PE_T=k{q_sq_T\over d}$$

Now this potential energy is hypothetical in this case: we didn't really put a charge at the star after all, we only imagined we did. But suppose I divided the hypothetical potential energy $PE_T$ by the hypothetical target charge $q_T$, to get $${PE_T\over q_T}=k{q_s\over d}$$ The right-hand side of this equation is made up entirely of non-hypothetical things: the constant $k$, the source charge $q_s$, and the distance from the star to the charge. Thus we can use this non-hypothetical quantity to describe the distortion of space at the star. We call this the electric potential V:

or, more specifically for a point charge,

(The potential is measured in units of joules per coulomb, which are called volts: 1 V=1 J/C.)

Another Example

As a more complicated example, consider three source charges on the corners of a square with side s as shown. Their total potential energy can be found by considering each of the three pairs of charges (represented by the three dashed blue lines), giving us $$PE_i=k{q_1q_2\over s}+k{q_1q_3\over s\sqrt2}+k{q_2q_3\over s}$$

Now suppose I want to know the electric potential $V$ at the fourth corner of the square, marked by the star. Place a hypothetical charge $q_T$ at the star, then the total potential energy of that configuration would be $$PE_f=\color{blue}{k{q_1q_2\over s}}+\color{blue}{k{q_1q_3\over s\sqrt2}}+\color{red}{k{q_1q_T\over s}}+\color{blue}{k{q_2q_3\over s}}+\color{red}{k{q_2q_T\over s\sqrt2}}+\color{red}{k{q_3q_T\over s}}$$ However, three of these terms, the terms represented by dotted blue lines, are the same as before. Only the terms with $q_T$ in them are new. Thus the hypothetical change in potential energy* $PE_T$ is due entirely to the terms which include $q_T$ in them: $$PE_T=\color{red}{k{q_1q_T\over s}+k{q_2q_T\over s\sqrt2}+k{q_3q_T\over s}} = q_T\left(k{q_1\over s}+k{q_2\over s\sqrt2}+k{q_3\over s}\right)$$ and thus the electric potential at the star is $$V={PE_T\over q_T}=k{q_1\over s}+k{q_2\over s\sqrt2}+k{q_3\over s}$$ Notice that if we calculated the potential at the star due to each point charge separately (e.g. $V_1=k{q_1\over s}$), and added them together, we would get the same result, so in future we don't need to go through the hassle of calculating the potential energy first.

Remember that there is nothing special about the star in the diagram above; there isn't actually a target charge there, we only pretended there was. We could have placed a target charge $q_T$ anywhere, found the change in potential energy $PE_T$, and still have calculated this quantity V at this new point. The potential of a set of source charges is defined everywhere in space.

Caveats

A few warnings to end things off with:

• The names electric potential energy and electric potential sound very similar and of course they are related, but they are very different. For one, the electric potential exists everywhere in space, whether there is anything there to feel it or not. Electric potential energy, on the other hand, isn't a property of space but a property of charges (or more precisely, groups of charges), and doesn't exist in a vacuum.
• The variable and the unit of electric potential are both "V", which can lead to some confusion: e.g. "V=5V". In this textbook, variables are always italicized like V, while units are never italicized, so while 5V represents "5 volts", $5V$ represents 5 times the electric potential. This may be more ambiguous in your own handwriting, so be careful.