A small source O of electromagnetic waves is placed some distance from a plane metal reflector M. A receiver R, moving between O and M along the line normal to the reflector, detects successive maximum and minimum readings on the meter joined to it. (a) explain why these readings are obtained. (b) calculate the frequency of the source O if the average distance between successive minima is 1.5 cm and the speed of electromagnetic waves in air= 3.0 x 10^8 m/s.

let w = 2 pi f

let k = 2 pi/wavelength L

wave moving right toward mirror
y = a sin (wt - kx)
wave moving left from mirror
y = a sin (wt + kx)
sum
y = a (sin(wt-kx) + sin(wt+kx))
then
sin(a+b) and sin(a-b)formula (trig)
y = a[sinwtcoskx+coswtsinkx+sinwtcoskx-coswtsinkx]

y = 2a [ sin wt cos kx ]
this is a standing wave, not traveling
it is node where cos kx = 0
that is when
kx = pi/2
kx = -pi/2
or
x = (pi/2)/(2pi/L)
x = -(pi/2)/(2pi/L)
distance between = pi L / 2 pi = L/2
given L/2 = .015 m
distance = rate * time
.015 = 3*10^8 T
get T
frequency = 1/T

(a) The readings obtained by the receiver R are due to the phenomenon of interference. When the electromagnetic waves from the source O reach the reflector M, they bounce off the metal surface and create reflected waves. When the reflected waves meet the original waves, they interfere with each other. This interference can be constructive or destructive, depending on the path difference between the original and reflected waves.

At certain locations along the line normal to the reflector, the path difference between the original and reflected waves is such that they reinforce each other, leading to constructive interference. This results in the receiver R detecting a maximum reading on the meter. At other locations, the path difference is such that the original and reflected waves cancel each other out, leading to destructive interference. This results in the receiver detecting a minimum reading on the meter.

As receiver R moves between O and M along the line normal to the reflector, the path difference gradually changes, leading to alternating maximum and minimum readings on the meter.

(b) To calculate the frequency of the source O, we can make use of the interferometer equation:
Δx = λ/2

Where:
Δx is the path difference between the original and reflected waves
λ is the wavelength of the electromagnetic waves

In this case, the average distance between successive minima is given as 1.5 cm. Since there is destructive interference at the minima, we know that the path difference between the waves is an odd multiple of half the wavelength. Therefore, we can write:

Δx = (2n + 1)λ/2

Rearranging the equation, we have:

λ = 2Δx / (2n + 1)

Given that the speed of electromagnetic waves in air is 3.0 x 10^8 m/s, we can now calculate the frequency (f) of the source O using the formula:

f = v / λ

Substituting the values:

f = (3.0 x 10^8 m/s) / (2Δx / (2n + 1))

f = (3.0 x 10^8 m/s) * (2n + 1) / (2Δx)

Substituting Δx = 1.5 cm = 0.015 m and assuming n = 0 (since the average distance between successive minima is given), we can calculate the frequency of the source O.