Coefficient of Potential & Capacitance | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

We know that inside a conductor there is no electric field and that the surface of a metal is an equipotential surface. Further, the tangential component of the electric field on the surface is zero leaving with only a normal component of the field on the surface. Our argument in arriving at these properties was largely intuitive and was based on our expectation that under electrostatic conditions, we expect the system of charges to stabilize and move around. Let us look at this problem from a mathematical stand point.

Consider surface of a conductor. Let us take the conductor surface to be in the xy plane. In the vacuum outside the conductor there are no free charges so that   Further as we are concerned with only electrostatic phenomena,  everywhere.

There are charges only on the surface of the conductor which would produce an electric field. The charges must be distributed in such a way that the fields they produce superpose so as to cancel the electric field inside the metal. So near the surface of the metal over a small a distance there is a change in the electric potential. Thus, if we take the outward normal to the surface of the conductor to be along the positive z direction, the z component of the electric field must be large near the surface.

If the surface is homogeneous,  must be finite and continuous. Since the curl of the electric field is zero, this means, for instance,

and since  is finite and continuous, so must be  and by symmetry  Since the field, and its tangential component, are zero inside the surface, the tangential components   must be zero on the surface.

Total charge on the surface of the conductor is obtained by integrating the surface charge density,

 which gives

Theorem : The potential function can be maximum only at the boundary of a region when the the electric field exists.

Proof : Suppose, the theorem is not true and the potential is maximum at a point P which does not lie on the boundary of a region of electric field. If this were so, we could enclose the point P by a small volume at every point of which the potential is less than that at P. Thus everywhere on this region   Thus the surface integral of this quantity  This contradicts the fact the according to Laplace’s equation this integral should be zero as the surface integral is equal to the amount of charge enclosed by the surface and away from the boundary there are no charges.

Energy of a Charged Conductor in Electric Field

We will calculate the energy that is stored in the field produced by a charged conductor. In doing so we will subtract the energy that is stored within the volume of the conductor itself. Let V’ denote all space around the conductor which excludes the conductor volume itself. The stored energy is given by

where we have used the identity

The second integral on the right gives zero because outside the conductor there being no charge,  The first integral can be converted to an integral over the surface using the divergence theorem. Since the volume excludes the conductor volume, the surface has two components, one at infinity, and the other at the surface of the conductor. The contribution to the surface integral from the surface at infinity is zero. We are then left with the integral over the surface of the conductor. Defining the normal direction on the conductor surface as being outward from the conductor surface, we observe that the boundary of the infinite region excluding the conductor volume has a normal direction which is opposite to the normal on the surface of the conductor.

We thus have

the integral being over the surface of the conductor itself.

If there are multiple conductors, we can extend the same argument and get the total energy stored in the field as

Where the potential has been taken out of the integral as the potential remains constant on each conductor surface.

The surface integral above, by Gauss’s theorem is the amount of charge  conductor. We thus have,

Since the field equations are linear, the potential and the charge are linear function of each other, the coefficient of proportionality can only depend on the shape of the conductor. Since,  for a single conductor, we have,

The coefficients   are known as the coefficients of capacitance and  are known as the coefficients of potential. In terms of these coefficients, the energy of a system of conductors can be written as

It can be shown that these coefficients are symmetric under interchange of indices. To illustrate this, let us assume that we have a collection of conductors and we add a small charge   only to the k-th conductor, keeping the charges on all other conductors the same. Let us look at the amount by which the total energy of the system changes. We have,

Where, in deriving the last step from its preceding step, we have changed the dummy summation index from I to j.

However, we can get an equivalent expression for the change in the energy from the expression  It follows that if the charge on the k-th conductor is increased by ΔQ_k, the energy would change by

(Remember that the factor of ½ in the expression for energy came in order to avoid double counting of interactions between pairs. Since we are only interested in change in potential energy due to an additional charge being given to the k-th conductor, the change in the potential energy is simply given by the product of the additional charge with the potential at the site of the k-th conductor.)

Comparing (1) and (2), we get

We can also show that the diagonal term   is the largest coefficient. Further, all coefficients are positive. 

Consider two conductors k and j. Suppose we bring in a unit positive charge from infinity to the k-th conductor, keeping all other conductors uncharged. By definition of the potential for a single conductor   is equal to the potential of the k-th conductor. Let us now bring the j-th conductor into the picture.

Since k-th conductor has a net positive charge, field lines Arising out of this conductor will either terminate at infinity, or terminate on the j-th conductor from which the field lines will go to infinity. Since Q_k is the only charge, the potential on the j-th conductor is   The potential on the k-th conductor is still   since  which gives,

In Lecture 5 we proves Green’s theorem according to which, given to scalar fields  in a volume V bounded by a surface S,

Let φ a potential field which arises from a volume charge distribution p in V and a surface charge distribution σ on S. Likewise, let ψ be a potential which arises from   Using Maxwell’s equation, we have  Let the electric field corresponding to φ and  respectively. We then have,

Thus

This is known as Green’s Reciprocity Theorem

Consider the volume to be bounded by conducting surfaces. Since the potential is constant on the conducting surfaces, we can write,

where the sum is taken over the surface of the conductors.

Example :

A point charge q is located at a distance r from a spherical conductor of radius R (<r). What is the potential on the surface of the sphere?

Since the only charge is the point charge at a distance r, we take volume charge density to be zero. We can consider the system as consisting of two conductors with the point charge being considered as a conductor with an infinitesimally small radius. If we put the charge Q_s = q on the surface of the sphere, the potential that it produces at the location of the charge at a distance r, is   If we use the reciprocity theorem, denoting the potential on the sphere when the charge is at the distance r from by ψ, 

which gives   (Note that this also follows from the definition of the coefficient of potential, Let the point charge be conductor 1 and the sphere as conductor 2.

 let the sphere be given a charge q. The potential at conductor 1 due to the charge q being given to the sphere is  However, the potential at the conductor 1 due to the charge q on the sphere is  By symmetry of the coefficient of potential,  The potential on the sphere when the charge on the point conductor is then,

We have here 

Example 2 :

Two conductors having capacitances C₁ and C₂ are placed apart at a distance r from each other. Calculate the capacitance of the pair.

Let us put a charge q on conductor 1 keeping the conductor 2 uncharged. The potential of the conductor 1 is then   and the potential on the conductor 2 due to the charge q at a distance r is  from the relation  Thus we have,

If we look at it as a matrix equation, we have  We can invert this equation to express the charges in terms of the potential on the two conductors,

The inverse of the matrix is easily found to be

Thus the coefficients of capacitances are given by,

Capacitors are devices which are used to store charges. Consider a collection of a system of conductors of which a pair hold equal and opposite charge  the others being neutral. Using the coefficients of potential, we can write the potential of these two conductors as

where φ_others is the potential of the uncharged conductors. The difference in potential of the two conductors is

The capacitance is thus given by

Force on a system of conductors 

Let us consider what happens when we displace a system of charges  If the force acting on the charges is   This work must be done by the charges at the cost of their energy,

We can then write

Suppose, instead, we have a system of conductors which are kept at some fixed potential   The energy of the system is

where   is the charge on the surface of the i-th conductor. It may be note that the charges q_i are unknown and must be obtained by solving Poisson’s equation.

Suppose an electric force displaces the system of conductors by   The work done by the force is   The problem at this stage is that we neither know the force nor know the work done. However, if we are armed with a knowledge of the total energy of the system, we can gain this information. The work can be done in two ways.

1. By a decrease in the stored electrostatic energy W^es  of the system : In this process the charges do not flow into or out of the system. The force is given by    with similar expressions for other components of the force.

2. By an increase in energy : If the potentials on the conductors are to be kept constant, work is done by drawing energy from the battery. This would result in an increase in the energy of the system and the force would be given by 

Example 3 : Force on a capacitor plate :

Consider the force exerted on one of the plates of a parallel plate capacitor. Imagine one of the plates as fixed and the other free to move. Assume that the plates are charged to   and the battery disconnected. In this situation, the charge on the plates is held constant. Now, the electrostatic energy is   The capacitance C depends on the instantaneous distance x of the movable plate from the fixed plate,

Thus

The force acting on the plate is  the negative sign shows that the force tends to reduce the distance between the plates.

Example 4 :

A parallel plate capacitor of width w and spacing d (<< w) between the plates is held fixed with both the plates being kept at a potential φ₀ with respect to the ground. A grounded conducting sheet of metal is partially inserted midway between the plates so that the left edge is at a distance x inside the capacitor plates. Calculate the force with which the sheet is being drawn inside.

Since both plates of the capacitor are at the same potential, the electric field between the plates is zero. However, when the grounded sheet is inserted, it creates an electric field in the region where the sheet is there. Since the spacing between the sheets and either plate is d/2, the electric field in the region is  Since the electric field is constant, the electrostatic energy

The integration is over the volume (=wxd) where the electric field exists. Thus

The positive sign indicates that the force tends to increase x i.e., draw the sheet inside.

Tutorial Assignment

(In solving these problems use the concepts developed in this lecture)

1. Consider two infinite grounded conductors separated by a distance of d. A point charge q is situated at a distance of a from the top conductor and b from the bottom conductor. What is the net charge induced on each of the conductors? How would the answer change if there is only the top conductor?

2. There is a cavity of irregular shape inside a spherical conductor of radius R. A charge q is placed inside the cavity being insulated from the walls of the cavity by an insulating prop. What is the potential at a distance r > R from the centre of the sphere?

3. Consider two infinite cylindrical conductors of radius a , their axes being parallel and separated from each other by a distance r > 2a. They carry equal and opposite charge of charge density +λ per unit length. Obtain an expression for the capacitance of the pair. What is the force exerted on one of the plates by the other if the potential difference between the plates is maintained at V?

4. Two concentric spherical conducting shells of radii R₁ and R₂ carry charge Q₁s and Q₂ respectively. Calculate the capacitance of the pair. What is the electrostatic energy stored in the pair?

Solutions to Tutorial Assignment

1. The original problem is shown in the figure to the left. Consider a second problem where the top plate is kept at a potential V₀ and the bottom plate is grounded. Both these situation have the same volume of space and are bounded by conducting surfaces.

In order to apply Green’s reciprocity theorem, we observe that that the charge and potential distribution of the two cases are as follows :

If the induced charges on the top plate on the top is Q_top that induced on the bottom plate is ,Q_bottom the volume charge densit   where the perpendicular distance z is measured from the bottom plate. The surface charges on the left are  for both the plates.

The charge density for the second case is zero both in volume and on the surfaces. The potentials are given by  for the bottom plate. The electric field being uniform between the plates, the potential at a distance z from the bottom plate is  Thus the potential at a distance b from the bottom is  Using,

expression on the left is zero   while that on the right is   Equating this to zero we get   By symmetry, one can write

If only the top plate remains, we can simulate the configuration by removing the bottom plate to infinity and get taking the limit  (We will see later that this is the solution obtained by method of images.)

2. Let us identify three conductors, viz., the sphere, which is conductor 1, the point conductor inside the irregular cavity the conductor 2 and a third point conductor 3 at the location P. From the definition of coefficients of potential we have,

These coefficients depend only on geometry and can therefore be determined by specifying charge distributions. Suppose, we put 

 However, in the absence of a charge in the cavity, these two potentials must be the same. Hence 

Next, let us put a charge q on the sphere leaving others uncharged, i.e.,  in this case, since the charge given to a conductor would only be symmetrically distributed on its outer surface, we would have   We then have  which gives,  Thus we have, 

Let us substitute these into the equation for

3. The potential due to a cylindrical conductor of radius of cross section a carrying a linear charge density λ at a distance ρ (>a) from the axis of the cylinder is is an arbitrary distance at which the potential is set to be zero (remember that for a cylindrical conductor we cannot take the reference point at infinite distance from the conductor). Using this it is straightforward to show that the capacitance of the pair is given by   If the a constant potential difference V is maintained between the conductors, the electrostatic energy is given by  so that the force is given by

4. When there is a charge Q₁ on the inner conductor and Q₂ on the outer conductor, the potential on the outer conductor is given by   The field between the spheres bein   one can easily calculate the potential on the inner sphere to be given b  Thus the potential matrix is given by

The capacitance matrix is given by

The electrostatic energy is given by

Self Assessment Quiz

1. Calculate the capacitance of two conductors carrying equal and opposite charges in terms of the coefficient of potential.

2. Three concentric shells of radii   respectively carry charges
Calculate the energy of the system.

3. Two concentric spherical shells of radii   are both grounded and a point charge q is placed at a distance r between them, being insulated from both the surfaces. Calculate the energy of the system charges.

4. Calculate the work done in doubling the separation between the plates of a parallel plate capacitor from d to 2d, if the potential difference between the plates remains φ.

Solutions to Self Assessment Quiz

2. The potential on the outermost conductor is  Potential on the central conductor is

The potential on the innermost conductor is

The matrix for the coefficient of potential is

The energy is given by, on substituting 

3. We use Green’s reciprocal theorem by constructing an artificial problem (figure to right) where the outer sphere is grounded but the inner sphere is maintained at a constant potential φ₀ without any charge being between them.

In the second case, the potential in the region between the spheres satisfies Laplace’s equation in variable r of spherical coordinates (as it can only depend on r) which has the solution   where A and B are constants. Substituting boundary conditions, the potential at r is given by

Since both spheres in given configuration is grounded, their potential is zero. Let Q_a and Q_b  be charges induced on the inner sphere and outer sphere respectively. We then have,
which can be solved to give

The charge induced on the outer sphere may be found by changing the boundary condition on the right to grounded inner sphere and the outer sphere being at a constant potential. It can also be inferred by realizing that the total induced charge should be -q.

4. The work done is calculated by the difference between initial and final energies. Since the potential difference between the plates is maintained at φ  If the initial separation is d, the capacitance is  and it becomes  when the separation is doubled. Work done is then