lec 02 2015 electromagnetic

Electromagnetic Field Theory

2nd Year EE Students

Prof. Dr. Magdi El-Saadawiwww.saadawi1.net

saadawi1@gmail.com

2014/2015

Chapter 1

VECTOR ALGEBRA(Continue)

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1.5. Vector Multiplication

Vectors may be multiplied by scalars: The magnitude of the vector changes, but its direction does not when the scalar is positive.

In case of vector multiplication:

the dot product (also called scalar product)

the cross product (also called vector product).

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1.5.1 The dot Product

Two vectors and are said to be orthogonal (or perpendicular) with each other if

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1.5.1 The dot Product

The dot product obeys the following identities:

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1.5.1 The dot Product

The most common application of the dot product is:

The mechanical work W, where a constant force F applied over a straight displacement L does an amount of work i.e.

Another example is the magnetic fields Φ, where

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1.5.2 The cross product

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1.5.2 The cross product

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1.5.2 The cross product

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1.5.2 The cross product

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1.5.2 The cross product

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1.5.2 The cross product

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1.6. The Gradient

The gradient of a scalar field is:

a vector field that lies in the direction for which the scalar field is changing most rapidly. The magnitude of the gradient is the greatest rate of change of the scalar field. (see figure 1.9 pp. 19)

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1.6. The Gradient

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1.6. The Gradient

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1.6. The Gradient

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1.7. Divergence of a vector and Divergence Theorem

The flux

Assume a vector field A, continuous in a region containing the smooth surface S, we define the surface integral of the flux of through S as:

Or

For a closed surface

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1.7. Divergence of a vector and Divergence Theorem

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If there are no sources within the boundary surface, thus the integral will get the value zero (Fig. b)

If there is a source (or sink) within the surface of integration, which generates new field lines the integral will get a value different from zero (Fig. a).

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1.7. Divergence of a vector and Divergence Theorem

The theorem of Gauss (Divergence theorem) is proved from the definition of the divergence and it enables to transform surface integrals into volume integrals as follows:

The volume integral about a specific flux from an element of volume V is equal to the flux thorough going from the closed surface S bounding this volume (V).

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Examples

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Examples

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Examples

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Examples

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