The Maxwell Equations in Vacuum

A. Maxwell Eqs expressed in INTEGRAL form

B. Maxwell Eqs. expressed in DIFFERENTIAL form

Line integral Surface integral

Operators Gradient Divergence Rotational Gauss' theorem Conversion of surface integral to volume integral Stoke's theorem Conversion of line integral to surface integral

Applied Optics Andres La Rosa Portland State University

A. The Maxwell Equations in Integral Form

The concept of line and surface integrals

=

= =

The First Maxwell Eq.

=

=

The Second Maxwell Eq. The Second Maxwell Eq.

=

The Third Maxwell Eq.

= =

The Fourth Maxwell Eq.

=

=

=

=

(2)

(1)

=

Fourth Maxwell Equation (1873)

Verification: Lets go back to our previous example and apply the "new" 4th ME and find out B using surface S1

No electric field E crossing the surface S1 so the 4th ME takes the form

(3)

=

=

What about if we choose the surface S2 ?

(4)

=

=

=

=

Fourth Maxwell Equation

Γ

Γ

More general:

= =

=

B. The Maxwell Equations in Differential Form

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=

= We see that, when V acts on a scalar field it gives a vector field.

(5)

Flux of a vector field across a surface S

Reference: R. Feynman, " The Feynman Lectures on Physics," Vol-II , Chapter 2 (Differential Calculus of Vector Fields;) Chapter 3 (Vector Integral Calculus).

Below we will generalize this result to the case in which the macroscopic volume is divided into multiple infinitesimal volumes

The Gauss' theorem

Flux through the cube of volume ΔV = Δx Δy Δz

(6)

First, let's calculate the flux across a surface containing an infinitesimal volume ΔV

(8)

On the other hand, let's generalize the result given in (5)

Using (6)

Using (5)

(7)

(9)

From (7) and (9)

Reference: R. Feynman, " The Feynman Lectures on Physics," Vol-II , Chapter 3 (Section 3-3 The Gauss theorem.)

Gauss theorem

Example. Expressing the charge conservation principle in differential form

V

Conservation of charge expressed in differential form.

Example. Expressing the SECOND Maxwell Eq. in differential form

FIRST Maxwell Eq. in differential form

Example. Expressing the FIRST Maxwell Eq.in differential form

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=

= =

=

SECOND Maxwell Eq. in differential form

=

=

The Stoke's Theorem

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Reference: R. Feynman, " The Feynman Lectures on Physics," Vol-II , Chapter 3 (Section 3-6 The Stokes theorem.)

Stoke's theorem

=

=

=

=

Third Maxwell Equation expressed in differential form

Example. Expressing the THIRD Maxwell Eq. in differential form

Example. Expressing the FOURTH Maxwell Eq. in differential form

This expression is valid for any arbitrary surface, including an infinitesimal rectangle. Thus,

Applying the Stoke's theorem

Assuming S is stationary

= -

=

=

=

= +

+

Fourth Maxwell Equation expressed in differential form

Since the expression above is valid for any arbitrary surface S, then,

= +