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Potentials and Gradients

Recall from MECHANICS that if we move a particle a vector distance $d\Vec{\ell}$ under the influence of a force $\Vec{F}$, that force does $dW = \Vec{F} \cdot d\Vec{\ell}$ worth of work on the particle - which appears as kinetic energy. Etc. If the force is due to the action of an electric field $\Vec{E}$ on a charge q, the work done is $dW = q \Vec{E} \cdot d\Vec{\ell}$. This work gets "stored up" as potential energy V as usual: dV = -dW. Just as we defined $\Vec{E}$ as the force per unit charge, we now define the ELECTRIC POTENTIAL $\phi$ to be the potential energy per unit charge, viz.

$$dV \; = \; q \, d\phi \qquad \hbox{\rm where} \qquad d\phi \; = \; - \Vec{E} \cdot d\Vec{\ell}$$ (17.11)

or, summing the contributions from all the infinitesimal elements $\Vec{\ell}$ of a finite path through space in the presence of electric fields,^17.9

$$\phi \; \equiv \; - \int \Vec{E} \cdot d\Vec{\ell}$$ (17.12)

When multiplied by q, $\phi$ gives the potential energy of the charge q in the electric field $\Vec{E}$.

Just as we quickly adapted our formulation of MECHANICS to use energy (potential and kinetic) as a starting point instead of force, in ${\cal E}$&${\cal M}$ we usually find it easier to start from $\phi(\Vec{r})$ as a function of position $(\Vec{r})$ and derive $\Vec{E}$ the same way we did in MECHANICS:

$$\Vec{E} \; \equiv \; - \Grad{\phi}$$ (17.13)

where, as before,^17.10

$$\Grad \; \equiv \; \hat{x} \, {\partial \over \partial x} \ . . . . . . \partial y} \; + \; \hat{z} \, {\partial \over \partial z}$$ (17.14)

The most important example is, of course, the electric potential due to a single "point charge" Q at the origin:

$$\phi (\Vec{r}) \; = \; k_E \, {Q \over r}$$ (17.15)

Note that $\phi(r) \to 0$ as $r \to 0$, as discussed in the previous footnote. This is a convenient convention. I will leave it as an exercise for the enthusiastic reader to show that

$$\Grad{\left(1 \over r \right)} = - {\hat{r} \over r^2} .$$

Electric potential is most commonly measured in volts (abbreviated V) after Count Volta, who made the first useful batteries. We often speak of the "voltage" of a battery or an appliance. [The latter does not ordinarily have any electric potential of its own, but it is designed to be powered by a certain "voltage." A light bulb would be a typical case in point.] The volt is actually such a familiar unit that eletric field is usually measured in the derivative unit, volts per meter (V/m). It really is time now to begin discussing units - what are those constants k_E and k_M, for instance? But first I have one last remark about potentials.

The electrostatic potential $\phi$ is often referred to as the SCALAR POTENTIAL, which immediately suggests that there must be such a thing as a VECTOR POTENTIAL too. Just so. The VECTOR POTENTIAL $\Vec{A}$ is used to calculate the magnetic field $\Vec{B}$ but not quite as simply as we get $\Vec{E}$ from $\Grad{\phi}$. In this case we have to take the "curl" of $\Vec{A}$ to get $\Vec{B}$:

$$\Vec{B} \; = \; \Curl{A} .$$ (17.16)

Never mind this now, but we will get back to it later.