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

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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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Further as we are concerned with only electrostatic phenomena, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) must be finite and continuous. Since the curl of the electric field is zero, this means, for instance,

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

and since Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is finite and continuous, so must be Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and by symmetry Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Since the field, and its tangential component, are zero inside the surface, the tangential components  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) must be zero on the surface.

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

Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which gives Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the surface integral of this quantity Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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

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

where we have used the identity

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

The second integral on the right gives zero because outside the conductor there being no charge, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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

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

the integral being over the surface of the conductor itself.

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

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

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

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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) conductor. We thus have,

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

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, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) for a single conductor, we have,

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

The coefficients  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) are known as the coefficients of capacitance and Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) are known as the coefficients of potential. In terms of these coefficients, the energy of a system of conductors can be written as

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

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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,

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

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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) It follows that if the charge on the k-th conductor is increased by ΔQk, the energy would change by

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

(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

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

We can also show that the diagonal term  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 Qk is the only charge, the potential on the j-th conductor is  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The potential on the k-th conductor is still  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) since Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which gives, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

In Lecture 5 we proves Green’s theorem according to which, given to scalar fields Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) in a volume V bounded by a surface S,

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

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Using Maxwell’s equation, we have Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Let the electric field corresponding to φ and Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) respectively. We then have,

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

Thus

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

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,

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

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 Qs = q on the surface of the sphere, the potential that it produces at the location of the charge at a distance r, is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)  If we use the reciprocity theorem, denoting the potential on the sphere when the charge is at the distance r from by ψ,  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

which gives  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) (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.

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

Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) let the sphere be given a charge q. The potential at conductor 1 due to the charge q being given to the sphere is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) However, the potential at the conductor 1 due to the charge q on the sphere is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) By symmetry of the coefficient of potential, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The potential on the sphere when the charge on the point conductor is then,

Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
We have here  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and the potential on the conductor 2 due to the charge q at a distance r is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) from the relation Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus we have,

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

If we look at it as a matrix equation, we have Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) We can invert this equation to express the charges in terms of the potential on the two conductors,

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

The inverse of the matrix is easily found to be

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

Thus the coefficients of capacitances are given by,

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

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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) the others being neutral. Using the coefficients of potential, we can write the potential of these two conductors as

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

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

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

The capacitance is thus given by

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

Force on a system of conductors 

Let us consider what happens when we displace a system of charges Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) If the force acting on the charges is  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) This work must be done by the charges at the cost of their energy,

Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
We can then write Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Suppose, instead, we have a system of conductors which are kept at some fixed potential  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The energy of the system is

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

where  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is the charge on the surface of the i-th conductor. It may be note that the charges qi are unknown and must be obtained by solving Poisson’s equation.

Suppose an electric force displaces the system of conductors by  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The work done by the force is  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 Wes  of the system : In this process the charges do not flow into or out of the system. The force is given by   Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and the battery disconnected. In this situation, the charge on the plates is held constant. Now, the electrostatic energy is  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The capacitance C depends on the instantaneous distance x of the movable plate from the fixed plate,

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

Thus

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

The force acting on the plate is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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.

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

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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Since the electric field is constant, the electrostatic energy

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

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

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

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.

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

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 Qtop that induced on the bottom plate is ,Qbottom the volume charge densit  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where the perpendicular distance z is measured from the bottom plate. The surface charges on the left are Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) for the bottom plate. The electric field being uniform between the plates, the potential at a distance z from the bottom plate is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the potential at a distance b from the bottom is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Using,

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

expression on the left is zero  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) while that on the right is  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Equating this to zero we get  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) By symmetry, one can write Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

If only the top plate remains, we can simulate the configuration by removing the bottom plate to infinity and get taking the limit Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) (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,

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

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

These coefficients depend only on geometry and can therefore be determined by specifying charge distributions. Suppose, we put  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) However, in the absence of a charge in the cavity, these two potentials must be the same. Hence  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

Next, let us put a charge q on the sphere leaving others uncharged, i.e., Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) in this case, since the charge given to a conductor would only be symmetrically distributed on its outer surface, we would have  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) We then have Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) which gives, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus we have, 

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

Let us substitute these into the equation for Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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 isCoefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) If the a constant potential difference V is maintained between the conductors, the electrostatic energy is given by Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) so that the force is given by

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

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) The field between the spheres bein  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) one can easily calculate the potential on the inner sphere to be given b Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Thus the potential matrix is given by

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

The capacitance matrix is given by

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

The electrostatic energy is given by

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

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) respectively carry charges Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
Calculate the energy of the system.

3. Two concentric spherical shells of radii  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) 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

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

2. The potential on the outermost conductor is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) Potential on the central conductor is

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

The potential on the innermost conductor is

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

The matrix for the coefficient of potential is

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

The energy is given by, on substituting  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

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

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.

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

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  Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) where A and B are constants. Substituting boundary conditions, the potential at r is given by

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

Since both spheres in given configuration is grounded, their potential is zero. Let Qa and Qb  be charges induced on the inner sphere and outer sphere respectively. We then have, Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
which can be solved to give

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

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 Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) If the initial separation is d, the capacitance is Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) and it becomes Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) when the separation is doubled. Work done is then Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

The document Coefficient of Potential & Capacitance - Notes | Study Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Electromagnetic Fields Theory (EMFT).
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