Microwave Lab Quiz

Lab Quiz: A microwave oven is in the back of the classroom. We have placed marshmallows in the microwave to make some measurements of the standing wave. Determine the frequency of the microwave. From this deduce a range in possible dimensions for microwaves including the smallest possible microwave. We also microwaved a cup of water. What is the total energy content of cavity? How many photons per second are oscillating in the microwave? What pressure do these photons exert on the side of the microwave?

Data:

wavelength = 12 cm (error = 1 cm)

dimensions: height = 23 cm, width = 35 cm, depth = 35 cm (error = 1 cm)

Microwaved cup:
mass = 100 g (error = 1 g)
time elapsed = 30 s (error = 1 s)
initial temperature = 20 degrees Celsius (error = 1 degree Celsius)
final temperature = 57 degrees Celsius (error = 1 degree Celsius)

Analysis:
frequency of microwave = (velocity of light)/(wavelength) = (3.00*10^8 m/s)/(0.12 cm)
                                        = 2.52*10^9 hz (error = 2.1*10^8 hz)

For this frequency the microwave could be any length which sets up a standing wave with a wavelength of 12 cm. The depth and height of the microwave would not matter given that the microwave radiation is emitted uniformly from one side. The width could be any value expressed as a multiple of 12 cm and some positive integer.

total energy for 30 seconds = (Tf - Ti)*(M)*(Cw) = (37)(100)(4.186) = 15500 J (error = 570 J)

Average Power = (total energy)/(time elapsed) = 517 W (error = 20 W)

Photons oscillating per second = (Average Power)/(energy of individual photon) = 517/(h*f)
h = planck's constant = 6.6261*10^(-34) kg*m^2/s, f = frequency = 2.52 Ghz

Photons oscillating per second = 3.12 * 10^26 (error = 2.6 * 10 ^25)

We assume that all surfaces of the microwave are perfectly reflective.
The pressure exerted on a perfectly reflective surface = I/c where I is the intensity given by the pontying vector

Assuming that pressure is exerted on all sides of the microwave equally we can approximate the electromagnetic intensity as the total power in the cavity divided by the surface area of the cavity.

Surface Area = 2WD + 2WH + 2DH = 2(0.23*0.35 + 0.23*0.35 + 0.35*0.35) = 0.5817 m^2
(error = 0.0375 m^2)

Intensity = Power/SA = 517/0.5817 = 894 W/m^2 (error = 93 W/m^2)

Pressure = I/c = 2.98*10^(-6) N/m^2 (error = 3.1 *10^(-7) N/m^2)