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ENE 325 Electromagnetic Fields and WavesPowerPoint Presentation

ENE 325 Electromagnetic Fields and Waves

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### ENE 325Electromagnetic Fields and Waves

Lecture 3 Gauss’s law and applications, Divergence, and Point Form of Gauss’s law

Review (1)

- Coulomb’s law
- Coulomb’s force
- electric field intensity
or

V/m

- Electric field intensity in different charge configurations
- infinite line charge
- ring charge
- surface charge

Outline

- Gauss’s law and applications
- Divergence and point form of Gauss’s law

Gauss’s law and applications

- “The net electric flux through any closed surface is equal to the total charge enclosed by that surface”.
- If we completely enclose a charge, then the net flux passing through the enclosing surface must be equal to the charge enclosed, Qenc.

Gauss’s law and applications

- The integral form of Gauss’s law:
- Gauss’s law is useful in finding the fields for problems that have a high degree of symmetry by following these steps:
- Determine what variables influence and what components of are present.
- Select an enclosing surface, Gaussian surface, whose surface vector is directed outward from the enclosed volume and is everywhere either tangential to or normal to

Gauss’s law and applications

- The enclosing surface must be selected in order for to be constant and to be able to pull it out of the integral.

Ex1 Determine from a charge Q located at the origin by using Gauss’s law.

1.

2. Select a Gaussian surface

3.Drat a fixed distance is constant and normal to a Gaussian surface, can be pulled out from the integral.

Ex2 Find at any point P (, , z) from an infinite length line of charge density L on the z-axis.

1. From symmetry,

2. Select a Gaussian surface with radius and length h.

3. D at a fixed distance is constant and normal to a Gaussian surface, can be pulled out from the integral. ant and normal to a Gaussian surface, can be pulled out from the integral.

Ex3 A parallel plate capacitor has surface charge +S located underneath a top plate and surface charge -S located on a bottom plate. Use Gauss’s law to find and between plates.

Ex4 Determine electric flux density for a coaxial cable.

Ex5 A point charge of 0.25 C is located at r = 0 and uniform surface charge densities are located as follows: 2 mC/m2 at r = 1 cm and -0.6 mC/m2 at r = 1.8 cm. Calculate at

- r = 0.5 cm
- r = 1.5 cm

c) r = 2.5 cm

Divergence and Point form of Gauss’s law(1)

- Divergence of a vector field at a particular point in space is a spatial derivative of the field indicating to what degree the field emanates from the point. Divergence is a scalar quantity that implies whether the point source contains a source or a sink of field.

where = volume differential element

Divergence and Point form of Gauss’s law(2)

or we can write in derivative form as

Del operator:

It is apparent that

therefore we can write a differential or a point form of Gauss’s law as

Divergence and Point form of Gauss’s law(3)

For a cylindrical coordinate:

For a spherical coordinate:

Physical example

The plunger moves up and

down indicating net movement

of molecules out and in,

respectively.

- positive indicates a source of flux. (positive charge)
- negative indicates a sink of flux. (negative charge)

An integral form of Gauss’s law can also be written as

Ex6Let . Determine

Ex7Let C/m2 for a radius r = 0 to r = 3 m in a cylindrical coordinate system and for r > 3 m. Determine a charge density at each location.

Ex8 Let in a cylindrical coordinate system. Determine both terms of the divergence theorem for a volume enclosed by r = 1 m, r = 2 m, z = 0 m, and z = 10 m.

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