## Waves and Optics, solutions

### Problem 1:  (C)

Speed of waves on a string

v = (T/m)½,   T = v2μ = 90000/100 = 900.

### Problem 2:  (A)

Sinusoidal traveling waves

### Problem 3:  (E)

Sound waves, Doppler shift

Let v = speed of sound.
f = f0(v - vobs)/(v - vs), where vobs and vs are not the speeds, but the components of the observer's and the source's velocity in the direction of the velocity of the sound reaching the observer.
f increases if the source and the observer approach each other and decreases if they recede from each other.
f =  f0 v/(v - vs) = [1/(1 - 0.9)]f0 = 10 kHz.

### Problem 4:  (E)

Images formed by mirrors

For the concave mirror:  1/xo + 1/xi = 1/f,  f = R/2 = positive.  If xo < f then xi is negative, we have a virtual image.

### Problem 5:  (D)

Sinusoidal traveling waves

v = ω/k = (5*1014/5*106) m/s = 108 m/s.  n = c/v = 3.

### Problem 6:  (C)

Interference

We have constructive interference, if the optical path length is the same or differs by an integer number of wavelength.
n1L1 = n2L2.  L1/L2 = (n2/n1).

### Problem 7:  (C)

Resolving power

Δθ ~ λ/D  = c/(fD) = 3*108/(30*109*10) = 10-3.
[Uncertainty principle:  ΔxΔpx ~ h,  Δpx/p = Δθ ~ h/(p Δx).  Δx = D, p = h/λ, Δ θ ~ λ/D.]

### Problem 8:  (E)

Polarization

For a polarizer Itransmitted = I0cos2θ.
A polarizer always absorbs half the intensity of unpolarized light.
I = ½I0cos2θ cos2(π/2 - θ) = (I0/8)sin2(2θ).

### Problem 9:  (C)

Single slit diffraction

w sinθ = nλ for the diffraction minima.

### Problem 10:  (B)

Thin film interference

When a light wave reflects from a medium with a larger index of refraction, then the phase shift of the reflected wave with respect to the incident wave is π (180o).  When a light wave reflects from a medium with a smaller index of refraction, then the phase shift of the reflected wave with respect to the incident wave is zero.
Here we have a 180o phase shift upon reflection on both interfaces of the coating.  For destructive interference we therefore need
2nd = (k + ½)λ, k = 0, 1, 2, ... .
d = (k + ½)*4000/(2*1.25) = (2k + 1)*800
(2k + 1) = odd integer