I) What are complex numbers?
A. Numbers that can be expressed as A + jB
where and A and B are real numbers.
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:
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
This means that differentiation and integration can
be (usually) replaced by multiplication and division!
IV) Anything else? Of course!
Thus a wave, for example, can be written as
We will explore this part more in quantum mechanics.
V) Do complex numbers have a geometric representation?
Well, that’s actually how they arose...
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.
… may be broken into its
… may be broken into its constituent parts
VI) How do I find the Magnitude of Complex Numbers…
Answer: Its easy, just multiply the number by its complex conjugate (and then take the square root).
But…what is a complex conjugate and how do I do it?
So to get the complex conjugate you change all the j’s to –j’s..that's it.
So…what’s 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 E1 = Eo, and let E2 = Eo
Then Enet = E1 + E2 = Eo+ Eo
Since the Intensity I =
Then we can find I by using E*E to get |Enet |2
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 = y1 + y2 = 2Aejkxcos (wt)
Where y1 = Aej(kx -wt) and y2= Aej(kx +wt )
show that if we choose the imaginary part of y for our “real” 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-jp/2
(a) A circuit has only an inductor of inductance L , and a Sinusoidal AC power supply with amplitude E.
Apply Kirchoff’s laws to determine a complex expression for the current in the circuit. Hint: Let the voltage source be represented by e = Eejwt and recall that the voltage drop across the inductor is –L dI/dt. Integrate (using the rules for integrating complex #’s) to find I.
(b) Show that the value of I you determined in part (a) can be written as
I = Ioejwt-jf where f= p/2, and Io = 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 Io = 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 = Ioej(wt + f) (Equation #1)
Now Consider for example a circuit with complex impedance
Z = 100ejp/6
If the power supply can be described the equation
e = 10ej(30pt)
Determine the value of Io and f.