I)
What are complex numbers?

A. Numbers
that can be expressed as A + jB

where
and A and B are
real numbers.

or

B. numbers
that can be expressed as where C and are real
numbers.

II) How are definitions A and B related?

By Euler's theorem:

=A+jB

Proof
:

III)
What good are they?

When you have to solve differential equations relating
harmonic (wave-like) functions they make

life
much easier.

In
particular, if

Thus,

This
means that differentiation and integration can

be
(usually) replaced by multiplication and division!

IV) Anything else? Of course!

If then

(look familiar?).

Thus
a wave, for example, can be written as

, etc.

We will explore this part more in quantum mechanics.

**V) **Do
complex numbers have a geometric representation?

Well, thatÕs actually how they arose...

let

Because
of Euler's formula we can represent the

real
and imaginary parts of Z as the components of a vector in the complex plane.

É may be broken into its
constituent parts

VI) How do I find the Magnitude of Complex NumbersÉ

Answer: Its easy, just multiply
the number by its __complex conjugate__ (and then take the square
root).

ButÉwhat is a complex conjugate and how do I do it?

OkayÉif ,

Then:

So to get the complex conjugate you change all the jÕs to ÐjÕs..that's it.

SoÉwhatÕs the point?

Well, if you multiply Z*Z, you get:

.
This means that you can find the magnitude of a complex number by taking
the square root of Z^{*}Z.

Example: How can you use complex numbers to derive double slit intensity distribution?

A: Let E_{1} = E_{o}_{}, and let E_{2} = E_{o}_{}

Then
E_{net } = E_{1} +
E_{2} = E_{o}_{}+ E_{o}_{}

Since the Intensity I =

Then
we can find I by using E^{*}E
to get |E_{net} |^{2}

SoÉ

Problems Using Complex Numbers

Work
through the following examples below.
Turn your work in on paper, showing all your steps (even if the work
seems obvious).

Exercise
#1 Show that the formula for
adding two traveling waves of equal amplitude and frequency, but traveling in
opposite direction is:

y
= y_{1} + y_{2} = 2Ae^{jkx}cos (wt)

Where y_{1} = Ae^{j(kx
-}^{w}^{t)} and y_{2}= Ae^{j(kx +}^{w}^{t )}

show
that if we choose the imaginary part of y for our ÒrealÓ solution, then the
answer is the familiar standing wave equation as described in the text (HR
& W Chapters 17 and 18).

Exercise
#2 Show that the complex number

Z
= 1/jw

can
be written as

Z = (1/w)e^{-j}^{p/2}

Exercise
#3

(a) A circuit has only an inductor of inductance L , and a Sinusoidal AC
power supply with amplitude E.

Apply
KirchoffÕs laws to determine a complex expression for the current in the
circuit. Hint: Let the
voltage source be represented by
e = Ee^{j}^{w}^{t} and
recall that the voltage drop across the inductor is ÐL dI/dt. Integrate (using the rules for
integrating complex #Õs) to find I.

(b) Show that the value of I you
determined in part (a) can be written as

I
= I_{o}e^{j}^{w}^{t-j}^{f} where f= p/2, and I_{o}
= E/wL.

Exercise
#4 Repeat exercise #2 and #3 with
the inductor replaced by a capacitor...... only this time, show
that f= -p/2, and I_{o} = E wC.

Hint: this time the voltage drop on the
capacitor is -q/C, where q is equal to the time integral of I(t).

^{ }

^{ }

Exercise #5. Just like the total resistance in a circuit, the total impedance in a circuit can be used to determine the current flowing in or out of the power supply.

I = e/Z = I_{o}e^{j(}^{wt + }^{f)}
(Equation #1)

I

e Z

Now Consider for
example a circuit with complex impedance

Z = 100e^{j}^{p}^{/6}

^{ }

If
the power supply can be described the equation

e = 10e^{j(30}^{p}^{t)}

Determine
the value of I_{o} and f.