Graphics
Applications

This laboratory will expand your repetoire of graphics skills and bring back joyful memories of problems past (or maybe not).

1. Necessary Commands

In this laboratory you will need to draw more than one curve on a single plot. The commands you need to do that are demonstrated below. Notice the use of color to distinguish different curves.

f1[x_] := Sin[x];
f2[x_] := Cos[x];
plot1 = Plot[f1[x],{x,0,25},
 Frame->True,
 FrameLabel->{"x","f1(x)","",""},
 PlotStyle->RGBColor[1,0,0]
]
plot2 = Plot[f2[x],{x,0,25},
 Frame->True,
FrameLabel->{"x","f2(x)","",""},
PlotStyle->RGBColor[0, 0, 1]
 ]
Show[{plot1,plot2}, FrameLabel->{"x","f(x)","",""}]

2. Resonance in AC Circuits.

Recall from Physics 132 the relationship between the current amplitude and voltage amplitude for an alternating current circuit (see Chapter 33 in Physics for Scientists and Engineers by Serway and Beichner)

                                              (Subscript[V, 0]) = (Subscript[I, 0])Z

where (Subscript[V, 0]) and (Subscript[I, 0]) are the voltage and current amplitudes respectively. The quantity Z is the impedance defined by

                                            Z=(Sqrt[(R^2) + (((1/ωC - ωL))^2) ])

where R is the resistance, C is the capacitance, L is the inductance, and ω is the frequency. 

a. For L=1.0 mH, C=0.1 μF, and (Subscript[V, 0])=1V calculate the dependance of  (Subscript[I, 0]) on the frequency for three values of R, 10 Ω, 30 Ω, and 150 Ω, and make a figure with all three curves on it.

b. At what radial frequency, ω, is (Subscript[I, 0]) a maximum for each curve?

c. What is the signal-to-noise ratio (the ratio of the maximum (Subscript[I, 0]) to the average background) for each R value?

d. How does the full-width-half-maximum (FWHM) change for each value of R (i.e. linearly, quadratically, etc)?

3. Black-body radiation

The effort to develop the correct model for the emission of light from a hot, totally absorbing object (blackbody radiation) led ultimately to Planck's quantum hypothesis and the subsequent radical restructuring of  physics during this century. Planck was able to show that the energy density inside the blackbody is

                                           ρ(λ) = ((8πh)/c) (1/(λ^5)) (1/((e^(hc/λkT))-1))
                                           
where ρis the energy density, λ is the wavelength of the light emitted, h is Planck's constant, c is the speed of light, k is Boltzman's constant, and T is the temperature of the  blackbody. Notice these are all constants except for λ. Using this expression Planck was then able to derive a purely empirical law (at the time) known as Wein's displacement law:

                                          (Subscript[λ, max]) T= 0.2014 (hc/k)                    .
 
a. Derive Wein's displacement law from Planck's result by first setting dρ/dλ = 0. Then set 
hc/λkT = x and generate a transcendental equation in x.  You should obtain an expression of the general form below.

                                         f(x) = g(x)

b. Solve your equation graphically by writing a short code that will generate the appropriate curves (f(x) and g(x)) and draw them on a single plot using Mathematica. You should hand in your solution including any relevant figures. If you had done this in 1901, instead of Planck you might have received the Nobel Prize instead of him!