Electric Potential and Capacitance

Recall that a conservative force is one in which the work done on a particle that moves between two points depends only on the two points and not on the path followed. With this property, we were able to define a potential energy that could be associated with the force involved. Similarly, we can define a conservative vector field as a field that generates a conservative force. This allows us to introduce the concept of a potential for the field, in direct analogy to the potential energy for a force. Specifically, we define the potential, j, of a vector field, ƒ, by the relation

(17)

Notice that the potential is a scalar quantity. Instead of three (or more) equations to solve, there is only one. It is this fact that makes the potential easier to work with in most calculations. Also, just as with potential energy, the potential of a field is not an exact quantity. Instead, it only has physical meaning when changes in the potential are considered. In fact, the field potential is related to the potential energy in exactly the same way the field is related to the force (up to a change in sign), i.e., by

(18)

where q₀ is the charge associated with the field under consideration.

Equipotential Surfaces

When DW = 0, we have a special situation. By equation (18), we have that Dj = 0. The locus of points which have Dj = 0 is called an equipotential surface. All equipotential surfaces are at right angles to the field lines everywhere. This fact makes it very easy to find the equipotential surfaces once the field lines are known (or vice versa).

Example:
Find the equipotential surfaces for two charges of equal magnitude but opposite signs separated by a distance a.

When dealing with electromagnetism, it is common to denote the electric potential by the symbol, V, which has units of volts. In terms of more fundamental units, a volt is defined as

1 volt = 1 joule / coulomb.

Since electromagnetism is a long range force, it is common to take the reference point, V_a, to be located at infinity, so that V_a = 0. When this choice is made, then DW is just the work required to move a test charge in from infinity to the point in question. Notice though, that there is no requirement that V_a be at infinity; occasionally, the inherent symmetry of the problem will dictate another choice for the location of V_a. An example of this will be given later.
Let us now consider a single point charge. What is the potential due to this charge? Recall that the electric field for the charge is given by

Then we find, for a point charge

(19)

If we have a collection of discrete charges. then the potential for the system is just the sum of the individual potentials:

(20)

Example:
What is the potential of an electric dipole?

Assume that the two charges have a magnitude of q and are separated by a distance 2a. For simplicity, place the center of the dipole at the origin. Then for a point at a distance r from the origin and at an angle q from the dipole direction, we have

where

and

In the limit r >> 2a, this reduces to

(21)

where p = 2aq is the electric dipole moment. Notice that the dipole is strongest along it's axis of orientation (q = 0, p) for any radius.

The electric dipole moment is found in many molecules and atoms. It is formed when the distribution of the charge is not symmetric about the physical distribution of the object. In these cases, there is an excess of positive charge at one side, and negative charge at the other. When the object is given a dipole moment by an external electric field, then the dipole is known as an induced electric dipole moment. When the external field is removed, the induced dipole moment will also disappear.

Example:
What is the potential of a spherical shell of radius R with a charge Q?

For a spherical charge density, the electric field is given by

Since we are working with a shell instead of a solid sphere, this is split into three sections: r > R, r = R, and r < R. Let us look at each one separately.

Case 1: r > R.
In this case, the electric field is given by

Since this is the electric field of a point charge, we have already found the potential. It is

Case 2: r < R.
In this case, the electric field is zero (all of the charge is on the surface of the shell). So we see that

V_b - V_a = 0 V_b = V_a = const

where the constant is yet to be determined.

Case 3: r = R.
This case is what is called a boundary condition. Since V is a continuous function, the values of V(r>R) and V(r<R) must be equal at r = R. Looking at V(r>R), we see that, at r = R, we have

Thus, we see that the unknown constant for case 2 is just

Summing all this up, we have that, for a spherical shell, the potential is given by

(22)

In this example we can see an interesting phenomenon. We can write Q as a density times a volume. As the volume is decreased, the density will increase. This causes a buildup of large potentials and fields at sharp points and corners. If the charge buildup is large enough, then the electric field generated by the charge will cause the air around it to ionize. This effect is known as a corona discharge and the best known example of it is lightning during a thunderstorm.

Example:
What is the potential of an infinite line charge?

The change in the voltage is given by

DV = - E · Dl.

The electric field of an infinite line charge is given by

(23)

where l is the charge per unit length. Taking Dl = Dr r and summing over all the Dr, we get

where V_a is our reference potential. Recall that earlier I said that it is common to take the reference potential, V_a, to be located at infinity, so that V_a = 0. In this case, if we take r = ¥, instead of getting V_a = 0 (as we would want) we would find V_a = ¥. Thus to have V_a = 0, we must take some arbitrary distance r = a. Then the potential becomes

(24)

Energy of the Electric Potential

Since the electric potential is defined in terms of the potential energy, we can use our knowledge of the electric potential to determine the energy stored in the electric field. In general this is given by

Example:
How much energy is required to move an electron through a potential difference of 1 V?

Using the relation between energy and electric potential we have

This combination shows up so frequently in particle physics that it has been given it's own designation, the electron volt, or eV. We frequently talk about particles with kinetic energies in the thousands, millions, billions or even trillion eV, so these are denoted keV, MeV, GeV and TeV respectively.

Capacitance

Once we have defined the electric potential, the concept of capacitance follows naturally. The capacitance of an isolated conductor is defined to be the ratio of the charge to the potential of the conductor. In terms of symbols, this is written

(25)

The SI unit of capacitance is the farad. From (25) it can be seen that one farad is equal to a coulomb per volt. Notice that, although the capacitance is defined in terms of the potential, the potential will always be proportional to the charge on the conductor, so the capacitance will depend only on the shape of the conductor.

Example:
For a spherical shell of radius R with a charge Q, we saw that the potential is given by

Therefore, the capacitance of a spherical shell is

(26)

A capacitor is a system of two conductors carrying equal and opposite charge. Note that they do not have to have the same shape, but must have the same charge. In terms of field lines, this means that all the field lines that originate on the positive conductor must end on the negative one. Thus, the overall charge of a capacitor is zero. Recall that the electric potential is defined only up to an additive constant. This allows us to define the potential of one of the conductors to be zero. Then, the potential that is used in (25) is the potential difference between the two conductors.
In practice, the capacitance is not difficult to calculate once the potential difference is known. The problems arise in trying to calculate the potential difference for complex arrangements of the conductors. As we shall see, even for simple arrangements, this calculation depends upon the shapes of the various conductors and the distances between them.

Example:
Consider two parallel conducting plates, each with a surface area of A, separated by a distance d. Let one plate carry a charge q and the other plate have a charge -q. This is known as a parallel plate capacitor.

To find the capacitance we have to find the electric field. For an infinite sheet of charge, the electric field is given by

(27)

and points from the positive plate towards the negative one. Here s is the charge per unit area. For a finite plate, (if we neglect the effect of the edges) we can replace s by q/A. Notice that this electric field exists only in between the two plates; outside them the field is zero. This can be seen by considering the field lines. Only inside the plates do the field lines point in the same direction. Once we have the electric field, we can find the potential difference between the plates. It is

(28)

where d is the distance between the two plates. Finally, once we have the potential, we can find the capacitance from (25). Thus,

(29)

Notice that the capacitance is independent of charge, proportional to the size of the conductors and inversely proportional to the distance between them.

Energy Stored in a Capacitor

Physically, a capacitor is used to store electrical energy in a circuit for use at a later time. How much energy is stored in a capacitor? We can calculate this by using the definition of capacitance and the relation between potential energy and electrical potential. Capacitance is defined as

where V is the voltage difference between the two pieces of the capacitor. Since the voltage is related to the potential energy by

we can calculate the energy stored by replacing the change in electric potential with the average voltage established between the capacitor plates:

Thus, the energy stored in the capacitor is just

or, upon using the fact that q = CV,

(30)

Capacitors in Series and Parallel

In electrical circuits, we often encounter combinations of capacitors and need to know what their combined effect is. The two most common combinations are capacitors in series (one after the other) and in parallel (linked side by side). In fact, all other combinations can be reduced to these combinations.
Let us look at a set of capacitors hooked up in parallel. For simplicity, let us assume that we are working with parallel plate capacitors. The potential on the top plate of each of the capacitors is the same, as is the potential on the bottom plate. Thus, the potential difference between the two plates is the same for each capacitor. If the charge on the first capacitor is Q₁ and the charge on the second one is Q₂, then the total charge on both capacitors is Q = Q₁ + Q₂ = C₁V + C₂V = (C₁ + C₂)V.

Forming the ratio Q/V, we see that the effective capacitance is given by

C_eff = C₁ + C₂.

(31)

Thus, we see that we can replace two capacitors in parallel by a single capacitor with an effective capacitance equal to the sum of the capacitance of the replaced capacitors. This result holds in general: We can always replace a chain of parallel capacitors by a single capacitor whose capacitance is the sum of the replaced capacitance. Symbolically, we write

(32)

If we have two capacitors hooked up in series, then the potential at the bottom of the first capacitor must equal the potential at the top of the second one, so the total potential difference across the two capacitors is the sum of the potential differences across each. The charge on each capacitor will be the same though. If the potential difference of the first capacitor is V₁ and the potential difference of the second capacitor is V₂ then

Forming the ratio 1/C_eff = V/Q, we see that the effective capacitance is given by

(33)

The effective capacitance of two capacitors in series is less than that of either one of the individual capacitors. In general we have, We can always replace a chain of series capacitors with a capacitor for which the reciprocal of the effective capacitance is the sum of the reciprocals of the individual capacitors. Symbolically, we write

(34)

Dielectrics

A nonconducting material, e.g,, glass or wood, is called a dielectric. If the space between the two conductors of a capacitor is completely filled with a dielectric, the capacitance increases by a factor c which is characteristic of the dielectric and called the dielectric constant.
Suppose a capacitor of capacitance C₀ is connected to a battery which charges it to a potential difference V₀ by placing a charge Q₀ = C₀V₀ on the plates. If the battery in now disconnected and a dielectric is inserted, filling the space between the plates, the potential difference decreases to a new value

(35)

Since the original charge Q₀ is still on the plates, the new capacitance is

(36)

Since the potential difference between the plates of a parallel plate capacitor is just the electric field between the plates times the separation d, the effect of the dielectric is to decrease the electric field by the factor c. If E₀ is the original field without the dielectric, the new field E is

E = E₀/c.

(37)

Thus, the effect of a dielectric is to replace the original potential, capacitance and electric field with (35), (36) and (37) in all equations.
Another way of determining the dielectric is to look at another quantity, the permittivity, e. The permittivity is related to the dielectric constant by the formula

(38)