Sound and Wave - Wave phenomena

A string consists of two parts attached at x = 0. The right part of the string (x  0) has mass μ_r per unit length and the left part of the string (x  0) has mass  mass μ_l per unit length. The string tension is T. If a wave of unit amplitude travels along the left part of the string, as shown in the figure above, what is the amplitude of the wave that is transmitted to the right part of the string?

A. 1
B. 

C. 

D. 

E. 0

(GR9677 #80)

Solution:

A. FALSE
Amplitude = 1, if  μ_l  = μ_r  but from the picture  μ_l ≠ μ_r
Since μ_l  μ_r suggests we expect  the transmitted amplitude will be less than 1

B. FALSE
The answer suggests that the transmitted amplitude will be bigger than 1, it should be less than 1
Let μ_l = 1, μ_r = 4
A = 2 / (1 + √¼) = 2/(3/2) = 4/3

C. TRUE
The answer suggests that the transmitted amplitude will be less than 1
Let μ_l = 1, μ_r = 4
A = 2(√¼) / (1 + √¼) = 1/(3/2) = 2/3

D. FALSE
The answer suggests that A = 0, if  μ_l  = μ_r

E. FALSE
Amplitude = 0 if  μ_r → ∞

Answer: C 

Nuclear & Particle Physics - X-rays

A beam of electrons is accelerated through a potential difference of 25 kV in an X-ray tube. The continuous X-ray spectrum emitted by the target of the tube will have a short wavelength limit of most nearly

A. 0.1 Å
B. 0.5 Å
C. 2 Å
D. 25 Å
E. 50 Å

(GR9277 #80)

Solution:

Energy: E = pc
Wavelength: λ = h/p = hc/E

E = 25 kV
h = 4.14 × 10⁻¹⁵ eV second
c = 3 × 10⁸ m/s
hc = 1.24 × 10⁻⁶ eVm

→ λ = ^{1.24 × 10−6}/_{25 × 10³}  = ½ × 10⁻¹⁰ m = 0.5 Å

Answer: B

Electromagnetism - Gauss' Law

Which of the following electric fields could exist in a finite region of space that contains no charges? (in these expressions, A is a constant, and i, j, and k are unit vectors pointing in the x, y, and z directions, respectively.)

A. A(2xyi − xzk)
B. A(−xyj + xzk)
C. A(xzi + xzj)
D. Axyz(i + j)
E. Axyzi

(GR8677 #80)

Solution:

Gauss’ Law for no charges:



Use method of elimination:

(A). FALSE
A(2xyi − xzk)
→

(B). TRUE
A(−xyj + xzk)
→

Answer: B

Special Relativity - Relativistic Addition of Velocity

A tube of water is traveling at ½c relative to the lab frame when a beam of light traveling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is 4/3)

A. ¹/₂ c
B. ²/₃ c
C. ⁵/₆ c
D. ¹⁰/₁₁ c
E. c

(GR0177 #80)

Solution:

Relativistic Addition of Velocities:



v = speed of the moving observer
u' = speed of object in the moving observer
u = speed of object in the moving observer relative to the rest observer

Given:
v = speed of water = ¹/₂ c
Index of refraction of water, n = ^c/_u'  = ⁴/₃
u' = speed of light in the water = ³/₄ c

u = speed of light in the water relative to the lab frame
= (¹/₂ c + ³/₄ c) / [1 + (¹/₂)(³/₄)]
= (⁵/₄ c)/(¹¹/8)
 = (⁵/₄ c)(⁸/11)
 = ¹⁰/₁₁ c

Answer: D