Thermal Physics - Power

A 100-watt electric heater element is placed in a pan containing one liter of water. Although the heating element is on for a long time, the water, though close to boiling does not boil. When the heating element is removed, approximately how long will it take the water to cool by 1 degree Celsius? (Assume that the specific heat for water is 4.2 kJ/kg^oC)

A. 20 s
B. 40 s
C. 60 s
D. 130 s
E. 200 s

(GR9677 #13)

Solution:

P = 100 W
V = 1 L = 1 m³ → m = 1 kg (STP)
ΔT = 1^oC
c = 4.2 kJ/kg^oC

Q = mcΔT = Pt
1 × 4200 × 1 = 100t
t = 42 s

Answer: B

Electromagnetism - Maxwell's Equation

Listed below are Maxwell’s equations of electromagnetism. If magnetic monopoles exist, which of the these equations would be INCORRECT?

I. Curl H = J + ∂D/∂t
II. Curl E = −∂B/∂t
III. div D = ρ
IV. div B = 0

A. IV only
B. I and II
C. I and III
D. II and IV
E. III and IV

(GR9277 #13)

Solution:

Gauss' law of magnetism: ∇ · B = 0
→ There is no magnetic monopole
If there is one, ∇ · B ≠ 0

Faraday's law of induction: ∇ × E = −∂B/∂t
→ if there is magnetic monopole, ∇ × E = −∂B/∂t + (current of magnetic monopole)

Answer: D

Optics - Interference

Two coherent sources of visible monochromatic light form an interference pattern on a screen. If the relative phase of the source is varied from 0 to 2π at a frequency of 500 hertz, which of the following best describes the effect, if any, on the interference pattern?

1. It is unaffected because the frequency of the phase change is very small compared to the frequency of visible light.
2. It is unaffected because the frequency of the phase change is an integral multiple of π.
3. It is destroyed except when the phase difference is 0 to π.
4. It is destroyed for all phase differences because the monochromaticity of the sources is destroyed.
5. It is not destroyed but simply shifts positions at a rate too rapid to be detected by the eye.

(GR8677 #13)

Solution:

Interference pattern is observed only when the sources are coherent (both sources have identical λ, f, and phase relationship).

The problem states that the light sources are coherent, thus the interference patterns are observed, not destroyed → (C) and (D) are FALSE.

Phase change will affect the constructive and destructive patterns of interference → (A) and (B) are FALSE.

Note: 500 Hz or 500/second is too rapid for human eye which only can perceive 60 Hz to 80 Hz flickering light.

Answer: E

Optics - Telescope

Two stars are separated by an angle of 3 × 10⁻⁵ radians. What is the diameter of the smallest telescope can resolve the two stars using visible light (λ = 600 nanometers)? (ignore any effect due to Earth’s atmosphere)

A. 1 mm
B. 2.5 cm
C. 10 cm
D. 2.5 m
E. 10 m

(GR0177 #13)

Solution:

Rayleigh telescope resolution limit: sin θ = 1.22 λ/D

Since θ ≪, assume sin θ ~ θ

D = ^1.22λ/_θ 

λ = 600 nanometers = 600 × 10⁻⁹ m

D = ^{1.22(600×10−9)}/_{(3×10⁻⁵)} = 2.44 × 10^-2 m ≈  2.5 cm

Answer: B