elc 2050: fields & wave propagationclick to edit elc 2050 © mohamed o. ashour 2018 slide 6...

ELC 2050: Fields & Wave Propagation

Department of Electronics and Electrical Communications Engineering

Introduced By:Eng. Mohamed Ossama Ashour

E-mail: vert4231@gmail.comFirst term 20-21

Agenda

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• Uniqueness theorem

• Images method

• Point Charge near an infinite conducting Grounded Plate problem

• Images method for 3D Problems (For reading only)

• Useful results for images method (For reading only)

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Uniqueness theorem

• What are the conditions for the solution of the above problem to be unique?

• Is it possible to have two different complete solutions for the same problem?

Certain charge distribution

The same charge distribution as (a)

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Uniqueness theorem

• A solution of Poisson's equation (of which Laplace's equation is a special case) that satisfies the given boundary conditions is a unique solution. This statement is called the uniqueness theorem.

• The implication of the uniqueness theorem is that the potential in a volume is uniquely determined if (a) the charge density throughout the region, and (b) the value of 𝑉 on all boundaries, are specified.

• The proof for the theorem can be found here or on pages 158,159 in the textbook “David K. Cheng, Field and Wave Electromagnetics”.

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Images method

• There is a class of electrostatic problems (the ones with objects that disturb the medium causing a discontinuity on it) with boundary conditions that appear to be difficult to satisfy if the governing Poisson’s (Laplace’s) equation is to be solved directly.

• But the conditions on the bounding surfaces in these problems can be set up by appropriate image (equivalent) charges, and the potential distributions can then be determined in a straightforward manner.

• This method of replacing bounding surfaces by appropriate image charges is called the method of images.

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Images method

Rules

1. To find a solution by image in a region, don’t change anything in that one while change other region(s).

2. Satisfy the same BCs in both the original & the equivalent problems.3. Apply the charge conservation principle in both the original & the

equivalent problems.

BCs to be satisfied &

• Note: Number of equivalent problems is equal to the number of dielectrics given in the problem.

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Example for the usage of images method

Point Charge near an infinite conducting Grounded Plate.

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• Note: Solution is only valid on the LHS region.

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Sheet 2

Assignment 3

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Sheet 2 Answers

7)

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Sheet 2 Answers

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Sheet 2 Answers

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Sheet 2

Assignment 4

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Sheet 2 Answers

8)

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Sheet 2 Answers

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Sheet 2 Answers

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Sheet 2

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Sheet 2 Answers

9)

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Sheet 2 Answers

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Sheet 2 Answers

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Images method for 3D Problems (For reading only)

Assignment 1

Problem 1

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Images method for 3D Problems (For reading only)

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Images method for 3D Problems (For reading only)

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Useful results for images method (For reading only)

Point Charge Outside a Conducting Grounded Sphere.

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• Note: Solution is only valid outside the Conducting Sphere.

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Images method for 3D Problems (For reading only)

Problem 2

Assignment 2

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Images method for 3D Problems (For reading only)

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Images method for 3D Problems (For reading only)

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Useful results for images method (For reading only)

Point Charge inside a Conducting Grounded Spherical Shell.

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• Note: Solution is only valid inside the Spherical Shell.