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Electric Potential

As for all basic forces of nature, the electrostatic force is a conservative force. Therefore the work done by the electrostatic force is independent of the path taken and we can define an electrostatic potential energy that can help us to implement energy conservation in mechanical problems. U (r ) = - F(r ') dr '

r r0

In words, the electrostatic potential energy equals the work done by the electrostatic force in bringing the particle from the location of interest (r) to the reference location (r0). Knowing the potential energy allows us to derive the electrostatic force. We simply calculate the gradient of U(r) to obtain: F(r ) = - U r

The gradient is a vector that points in the direction of steepest ascent. Electric Potential Energy for Point Charge: As an example lets calculate the electric potential energy associated with a point charge q in the vicinity of another point charge Q. Symmetry tells us that the potential can only depend on the distance between the two charged objects. We choose infinite separation to be the point of reference. Then at a separation, r the electric potential is U (r ) = -

r

Qq Qq dr = 2 4 0 r 4 0 r

We notice that the electrical potential energy is equivalent to the gravitational potential energy though with masses substituted by charges. We check that we recover Coulombs law when calculate the gradient of the potential energy: F(r ) = - Qq Qq ^ 4 r = 4 r 2 r r 0 0

Definition of the Electric Potential Just as we defined an electric field from the electrostatic force, we define an electric potential from the electric potential energy by dividing out a "test charge". The idea is to associate an electric potential with a charge distribution. We imagine measuring this Collin Leslie Broholm Page 1 02/07/00 1

potential by measuring the electric potential energy for a test charge in the vicinity of this charge distribution, then divide this function with the magnitude of the test charge to obtain a function that characterizes the charge distribution. V (r ) = U (r ) q

We see that the units for the electrical potential is J/C. We use this unit so often in electromagnetism that we have given it a special name Volt or in short V. Using this new unit we see that the unit for electric field N/C can also be written as V/m. Calculating the Electric Potential A charge distribution yields a spatially varying electric potential. We can calculate the potential from a general charge distribution starting from the expression for the electric potential associated with a point charge. This is trivially derived from the expression for the electrical potential energy associated with a point charge by dividing with the test charge:

V (r ) = Q 4 0 r

Note that a positive point charge creates a positive potential. The expression looks a lot like the expression for the electric field from a point charge but there are crucial differences that could be overlooked at first glance: · · The electric potential is a scalar the electric field is a vector r enters to the first power in the denominator for the electrical potential. It enters as r2 in the denominator of the expression for the electric field from a point charge.

The superposition principle allows generalization from this simple result. If we have several discrete point charges, the electrical potential is calculated via: V (r ) =

i

Qi 4 0 ri

If we have a continuous charge distribution, the electric potential is derived through a suitable integration:

V (r ) =

(r ')d 2 r 1 V (r ) = 4 0 S ' R

V (r ) =

(r ')d 3 r 1 4 0 V ' R

for volume charge distribution

for surface charge distribution

(r ')dr 1 4 0 L ' R

for line charge distribution Page 2 02/07/00 2

Collin Leslie Broholm

You will notice the similarity of these expressions with those we used to calculate the electric field form continuous charge distributions. On close inspection, you will notice that the expressions for V(r) are significantly simpler than those for E(r) because the former is a scalar while the latter is a vector. This is part of the reason that we care to introduce and work with the electrical potential. Deriving V from E If we have E(r) and we want V(r), we use the following expression that follows from the way we derive a potential energy from a force: V (r ) = - E(r ) dr

r r0

The integration goes from the point of reference (r0) where the electric potential by definition is zero to the point (r) where we want to calculate the electric potential. As an example, we calculate the potential difference between Two charged metallic plates: Assume the charge on the left and right plates is +Q and -Q respectively. The field outside the plates must vanish (an application of Gauss law will show this). Therefore the charge density on the inner surfaces of the plates is =Q/A and from Gauss law, we easily derive that the electric field between the plates is E(r ) =

^ z 0

where the unit vector points from the positively charged plate to the negatively charged plate. The electric potential between the plates is V (r ) = - E(r ) dr = V (r0 ) -

r r0

z 0

+

-

The total electric potential difference between the plates is V = V+ - V- =

z

d 0

d

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V is Constant in a Metal In electrostatics, there can be no electric field in the interior of a conductor. If there were, then charge would flow and we would not yet have achieved the electrostatic condition. We used this result to derive the fact that charge lies on the surface of a conductor from Gauss law. From this result comes also an interesting result about the electric potential. Because V (r ) = - E(r ) dr

r r0

We can say the any point within a volume of space where the electric field vanishes has the same electric potential. Think about it for a while and this result should not be a surprise at all. If the electric field vanishes there can be no force on any charged particle which in turn means that there can be no electrical potential energy difference and hence no electric potential difference. This result makes it clear why the interior of a metallic vehicle is relatively safe from lightning in a thunderstorm. Certainly if the car formed a complete closed shell (no windows) then the shell and the interior and the passengers would be part of the same equipotential surface and hence there would be no danger within the vehicle from electric field differences. Even if we allow for windows it turns out that the vehicle offers relative safety as long as the windows are small compared to a characteristic electromagnetic wavelength. Once we go to an open vehicle, it cannot be guaranteed that the passengers will be safe on an equipotential surface. If we go to a fiberglass Corvette the vehicle offers no safety at all. We can illustrate the ability of a metal to shield from electric fields by enclosing a small radio in a metallic cage. The radio signal is transmitted by setting up charge distributions on large antennae that in turn generate an electric potential that moves charge around on the antenna of the radio. This charge motion is detected by the electronics within the radio and converted to motion of the loudspeaker membrane and sound waves that we detect with our ears. However, if we enclose the radio in a metal cage then it becomes part of an equipotential volume. The charge on the antenna no longer experiences an electric force and the radio is no longer affected by the electric fields being generated by the antennae. The reason that the signal is shielded although there are openings in the wire net, is that the openings are smaller than the wavelength for the electromagnetic waves. This is a point that we will return to later. Deriving E from V Just as we can derive a force from a potential energy, we can derive the electric field vector from the electric potential: F(r ) = - U r E(r ) = - V r 02/07/00 4

Collin Leslie Broholm

Page 4

As an example of an application of this, we calculate the electric field around conducting spheres held at the same potential but have different radius. We know that a charged sphere is indistinguishable from a point charge at the center of the sphere throughout the region of space outside the sphere. Thus, the electric potential outside a sphere must also be the same as that around a point charge: V (r ) = Q 4 0 r

If we charge two spheres to electric potential Vs then the charge on the surface of the sphere is evidently given by Q = 4 0 RV Where R is the radius of the sphere. The larger sphere has the larger charge. This seems sensible. However, if we instead look at the charge density on the surface of the sphere we find: 4 0 RV 0V Q = = = 2 R 4R 4R 2 The expression implies that the surface charge density is greater on the smaller sphere. Thus, it should not be a surprise that the electric field is also largest around the smaller sphere: E(r ) = - V Q Q ^ ^ V (r ) = - = r= sr 2 r r 4 0 r 4 0 r r

Specifically we see that the magnitude of the electric field on the surface of spheres held at the same potential is inversely proportional to the radius of the spheres.

We check this result by a few demonstration experiments: 1. Draw sparks between spheres charged to the same potential using the Wimshurst machine. Notice that the spark occurs between the smaller radius spheres. This is the experimental indication that the electric field is greater around the spheres with smaller radius. 2. We image the electric field pattern around a charged ovoid using the Wimshurst machine and fibers in an oil solution. The fibers concentrate around the region with small radius of curvature indicating again that the field is largest there. 3. Charge an ovoid with the van de Graff generator. Connect a metallic object to the large and small curvature part of the ovoid and see that the charge collected is greater when we connect to the part of the ovoid that has a smaller radius of curvature. Collin Leslie Broholm Page 5 02/07/00 5

Collin Leslie Broholm

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Electric Potential

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