Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) PDF Download

Laplace’s Equation in Spherical Coordinates :

We continue with our discussion of solutions of Laplace’s equation in spherical coordinates by giving some more examples.

Example 3 : A conducting sphere in a uniform electric field This is a classic problem. Let the sphere be placed in a uniform electric field in z direction Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Though we expect the field near the conductor to be modified, at large distances from the conductor, the field should be uniform and we have,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Clearly, we have azimuthal symmetry because of the sphere but the direction of the electric field will bring in polar angle dependence. As a result, the potential at large distances has the following form :

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where C is a constant. The conductor being a equipotential, the field lines strike its surface normally.

The lines of force near the sphere look as follows :

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

As shown here, there will be charges induced on the surface. On the surface, the potential is constant (it is a conductor) φ0. Since there are no sources, outside the sphere, the potential satisfies the Laplace’s equation. As there is azimuthal symmetry in the problem, the potential is given by,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Note that the potential expression cannot contain l = 0 term in the second term because a potential Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) implies a delta function source.

At large distances from the conductor, the above expression must match with  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) which we had obtained earlier. As all terms containing Bl vanish at large distances, we need to compare with the terms containing  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) are the only non-zero terms,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Thus,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

We will now use the boundary condition on the surface of the conductor, which is a constant, say  φ0  This implies

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Comparing both sides, we have, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)
The potential outside the sphere is thus given by,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The electric field is given by the negative gradient of the potential,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The charge density on the sphere is given by the normal component (i.e. radial component) of the electric field on the surface (r=R) and is given by

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The charge induced on the upper hemisphere is (recall that θ is measured from the north pole of the sphere as z direction points that way,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The charge on the lower hemisphere is Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) so that the total charge is zero, as expected.

Note that an uncharged sphere in an electric field modifies the potential by  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) You may recall that the potential due to a dipole placed at the origin is Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Thus the sphere behaves like a dipole of dipole moment Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

If the sphere had a charge Q, it would modify the potential by an additional Coulomb term.

Example 4 : Conducting hemispherical shells joined at the equator :

Consider two conducting hemispherical shells which are joined at the equator with negligible separation between them. The upper hemisphere is mained at a potential of  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) while the lower is maintained at Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) .We are required to find the potential within the sphere.

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Since the system gas azimuthal symmetry, we can expand the potential in terms of Legendre polynomials,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Using the argument we have given earlier, the relevant equation for the potential inside and outside the sphere are as follows :

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

We will determine a few of the coefficients in these expansions. To find the coefficients, we use the orthogonality property of the Legendre polynomials,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Using this we have,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Thus,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Since the potential is constant in two hemispheres, we split this integral from  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) corresponding to Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) corresponding to Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

We know that  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) is a polynomial in μ, containing only odd powers of μ if m is off and only even powers of μ if m is even, the degree of polynomial being m. It can be easily seen that for even values of m, the above integral vanishes and only m that give non zero value are those which are odd. We will calculate a few such coefficients.

Take m=1 for which  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) the integrals within the square bracket adds up to φ0, so that we get  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Take m=3 for which Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The integrals add up to  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) so that we get  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Laplace’s Equation in Cylindrical Coordinates :

In cylindrical coordinates , Laplace’s equation has the following form :

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

As before, we will attempt a separation of variables, by writing,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Substituting this into Laplace’s equation and dividing both sides of the equation by Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) we get,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where, as before, we have used the fact that the first two terms depend on p and θ while the third term depends on z alone. Here -k2 is a constant and we have not yet specified its nature. We can easily solve the z equation,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

which has the solution,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

(Remember that we have not specified k, if it turns out to be imaginary, our solution would be sine and cosine functions. In the following discussion we choose the hyperbolic form.) 

Let us now look at the equation for R and Q, Expanding the first term,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Multiplying throughout by p2, we rewrite this equation as

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Once again, we notice that the left hand side depends only on ρ while the right hand side depends on θ only. Thus we must have,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where v2 is an yet unspecified constant. The angle equation can be easily solved,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

which has the solution

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The domain of θ is [0:2π]. Since the potential must be single valued, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) which requires ν must be an integer.

The radial equation now reads

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The above can be written in a compact form by defining  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) in terms of which the equation reads,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

This equation is known as the Bessel equation and its solutions are known as Bessel Functions. We will not solve this equation but will point out the nature of its solutions. Being a second order equation, there are two independent solutions, known as Bessel functions of the first and the second kind. The first kind is usually referred to as Bessel functions whereas the second kind is also known as Neumann functions.

The Bessel functions of order ν is given by the power series,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Some of the limiting values of the function are as follows:
For  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) the function oscillates and has the form

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

When ν is an integer,  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) are not independent and are related by Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The variation of the Bessel function of some of the integral orders are given in the following figure.

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

It can be seen that other that the zeroth order Bessel function, all Bessel functions vanish at the origin. In addition the Bessel functions have zeros at different values of their argument. The following table lists the zeros of Bessel functions. In the following knm denotes the m-th zero of Bessel function of order n.

 

 Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)123
k0m02.4065.5208.654
k1m13.8327.01610.173
k2m25.1368.41711.620


Higher roots are approximately located at  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)
Bessel functions satisfy orthogonality condition,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Another usefulness of Bessel function lies in the fact that a piecewise continuous function f defined in [0,a] with f(a)=0 can be expanded as follows :

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Using orthogonality property of Bessel functions, we get,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The second kind of Bessel function, viz., the Neumann function, diverges at the origin and oscillates for larger values of its argument ,Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Like the Bessel function of the first kind, the Neumann functions are also not independent for integral values of n  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) variation of the Neumann function are as shown below.

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The solution of the radial equation can be written as

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

(If in the solution of z equation we had chosen k to be imaginary, the argument of the Bessel functions would be imaginary)
The complete solution is obtained by summing over all values of k and ν.

Special Case : A system with potential independent of z: In problems such as a long conductor, the symmetry of the problem makes the potential independent of z coordinates. In such cases, the problem is essentially two dimensional, the Laplace’s equation being,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

A separation of variables of the form  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) gives us, on expanding the first term,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The solution for Q(θ), is given by

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where, as before, singlevaluedness of the potential requires that n is an integer. The radial equation,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

has a solution

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

And for n=0

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Example : An uncharged cylinder in a uniform electric field.

Like the problem of sphere in an electric field, we will discuss how potential function for a uniform electric field is modified in the presence of a cylinder. Let us take the electric field directed along the x-axis and we further assume that the potential does not have any z dependence.

As before, at large distances, the potential is that corresponding to a constant electric field in the x direction, i.e  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The charges induced on the conductor surface produce their own potential which superpose with this potential to satisfy the boundary condition on the surface of the conductor. Since there is no z dependence, the general solution of the Laplace’s equation, as explained above is,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Here we have not taken n=0 term because its behavior is logarithmic which diverges at infinite distances. At large distances, the asymptotic behavior of this should match with the potential corresponding to the uniform field. Thus we choose the term proportional to  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Clearly  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) We need to determine the constant D1.  If the potential on the surface of the cylinder is taken to be zero, we must have for all angles θ, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Substituting these, we get,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The electric field is given by the gradient of the potential, and is

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The charge density on the surface of the conductor is the normal component of the electric field multiplied with  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Like in the case of the sphere, you can check that the total charge density is zero.

 

Tutorial Assignment

1. A cylindrical shell of radius R and length L has its top cap maintained at a constant potential φ0, its bottom cap and the curved surface are grounded. Obtain an expression for the potential within the cylinder.

2. A unit disk Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) has no sources of charge on it. The potential on the rim Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) is given by  2 sin 4θ, where θ is the polar angle. Obtain an expression for potential inside the disk.

3. Consider two hemispherical shells of radius R where the bottom half is kept at zero potential and the top half is maintained at a constant potential φ0, Obtain an expression for the potential inside the shell.

 

Solutions to Tutorial Assignment

1. Take the bottom cap in the x-y plane with its centre at the origin and the z axis along the axis of the cylinder. Since the potential at z=0 is zero, we take Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)  is within the cylinder, Neumann functions are excluded from the solution and we have,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Since the potential on the curved surface is zero irrespective of the azimuthal angle, we must v = 0. This gives,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The potential at  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) also vanishes, giving  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) This determines the values of  kn being given by the zeros of Bessel function of order zero. Finally, the boundary condition on the top cap,Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The coefficients An are determined by the orthogonal property of Bessel functions,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

using which we have, substituting Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)
Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Multiply both sides of eqn. (I) with  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) and integrate from 0 t0 R,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

2. Laplace’s equation in polar coordinates is  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) separation of variables Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

which have the solution  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Here we have used the singlevaluedness of the potential to conclude that λ is an integer. The solution of the radial equation is  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The constant D must be zero because the function must be well behaved at the origin. Likewise the solution of the radial equation for n=0 is a logarithmic function the coefficient of which also vanishes for the same reason. Thus the solution is of the form,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

We can determine the coefficients by the boundary condition given at r=1,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Clearly, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) all coefficients other than B4  are also zero and B4 = 2.

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

3. See example 4 of lecture notes. Inside the sphere the potential is given by

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Using the orthogonality property of the Legendre polynomial,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Using the fact that only one half of the sphere is maintained at constant potential while the other half is at zero potential, we have,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The integral can be looked up from standard tables,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Thus,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Using standard table for Legendre polynomials we have, 

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) etc. while all odd orders are zero. Thus,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Self Assessment Quiz

1. A conducting sphere of radius R is kept in an electrostatic field given by  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Obtain an expression for the potential in the region outside the sphere and obtain the charge density on the sphere

2. A unit disk  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) has no sources of charge on it. The electric field on the rim  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) is given by  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) is the polar angle. Obtain an expression for potential inside the disk.

3. (Hard Problem) A cylindrical conductor of infinite length has its curved surfaces divided into two equal halves, the part with polar angle  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) has a constant potential φ1 while the remaining half is at a potential φ2.Obtain an expression for the potential inside the cylinder. Verify your result by calculating the potential on the surface at   Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) and at  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

 

Solutions to Self Assessment Quiz

1. The potential corresponding to the electric field is  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Using the table for Spherical harmonics, the potential can be expressed as Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The general expression for solution of Laplace’s equation is

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The boundary conditions to be satisfied are

(i) For all  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) the potential is constant, which we take to be zero. This implies  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) for all m. This makes the potential expression to be

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

(ii) For large distances, Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) in this limit,, the second term for the expression for potential goes to zero and we are left with,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Comparing, we get only non-zero value of to be 2 and the corresponding m values as +2. Thus, the potential has the form

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The surface charge density is given by

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

2. The solution of Laplace’s equation on a unit disk is given by (see tutorial problem)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The electric field is given by the negative gradient of potential. Since the field is given to be radial,

 

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

This gives all coefficients other than B4 to be zero and  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The coefficient A0, however remains undetermined.

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

3. Because of symmetry along the z axis, the potential can only depend on the polar coordinates (p, θ) . By using a separation of variables, we can show, as shown in the lecture, the angular part Q(θ) =  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) The radial equation has the solution Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) we have ignored n=0 solution because it gives rise to logarithm in radial coordinate which diverges at origin. Thus the general solution for the potential is

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

where K is a constant. Further, the constant D must be zero otherwise the potential would diverge at p = 0. Let us now apply the boundary conditions, For p = R,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Integrate the first expression from  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) and the second expression from Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) and add. The integration over the second term on the right hand side of the two expressions is equal to an integration from 0 to 2π and the integral vanishes. We are left with

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Thus the potential expression at p = R becomes

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

To evaluate the coefficients An, multiply both sides by cos mθ and integrate from 0 to 2π,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Similarly, the first term on the right also gives zero.

We are left with

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

These integrals can be done by elementary methods,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

To determine the coefficients Bn, multiply both sides of (I) by sin mθ and integrate from 0 to 2π,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The first term on the right gives zero.
We are left with

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The integral can be easily done If m ≠ n,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

We have the final expression for the potential,

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

To verify that this is consistent with the given boundary conditions, let us evaluate the potential at  Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The infinite series has a value Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) Similarly, one can check that Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

The document Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE) is a part of Electronics and Communication Engineering (ECE) category.
All you need of Electronics and Communication Engineering (ECE) at this link: Electronics and Communication Engineering (ECE)

FAQs on Solutions to Laplace’s Equations - 3 - Electronics and Communication Engineering (ECE)

1. What is Laplace's equation and why is it important in electronics and communication engineering?
Ans. Laplace's equation is a second-order partial differential equation that describes the behavior of electric potentials, gravitational potentials, and fluid flows. In electronics and communication engineering, Laplace's equation is important because it helps in solving problems related to electromagnetic fields, signal propagation, and circuit analysis.
2. How can Laplace's equation be solved?
Ans. Laplace's equation can be solved using various methods such as separation of variables, Green's function, and numerical techniques like finite difference or finite element methods. The choice of method depends on the boundary conditions and the geometry of the problem.
3. What are the applications of Laplace's equation in electronics and communication engineering?
Ans. Laplace's equation finds applications in various areas of electronics and communication engineering. It is used to analyze the behavior of electric fields in capacitors and inductors, solve problems related to antenna radiation patterns, model the propagation of electromagnetic waves in transmission lines, and analyze the steady-state behavior of electronic circuits.
4. Can Laplace's equation be used to solve time-varying problems in electronics and communication engineering?
Ans. No, Laplace's equation is a steady-state equation that describes the behavior of systems in equilibrium. It cannot be directly used to solve time-varying problems. However, by introducing the concept of complex variables and using the Laplace transform, Laplace's equation can be extended to solve dynamic problems in the frequency domain.
5. Are there any limitations or assumptions associated with Laplace's equation in electronics and communication engineering?
Ans. Yes, there are certain limitations and assumptions associated with Laplace's equation. It assumes that the system is linear, homogeneous, and isotropic. It also assumes that the boundary conditions are well-defined and that the solution is unique. Additionally, Laplace's equation is not applicable to problems involving magnetic fields or moving charges, which require the use of the more general Maxwell's equations.
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