Nuclear & Particle Physics - Hydrogen Spectrum

In the spectrum of Hydrogen, what is the ratio of the longest wavelength in the Lyman series (n_f  = 1) to the longest wavelength in the Balmer series (n_f  = 2)?

A. 5/27
B. 1/3
C. 4/9
D. 3/2
E. 3

(GR9677 #09)

Solution:

Rydberg formula:



λ_vac = the wavelength of the light emitted in vacuum
R_H = Rydberg constant for Hydrogen
n₁ and n₂ are integers such that n₁  n₂

For the longest wavelength take n₂ = ∞

For Lyman-radiation (n₂ = ∞ → n₁ = 1):

1/λ_L = R_H (1/1 − 0) = R_H

For Balmer-radiation (n₂ = ∞ → n₁ = 2):

1/λ_B = R_H (¼ − 0) = ¼R_H

The Ratio:

^λ_L/_λB = (¹/_RH)(^R_H/₄) = ¼ ≈ 5/27

Answer: A

Electromagnetism - Conductor

A coaxial cable having radii a, b, and c carries equal and opposite currents of magnitude i on the inner and outer conductors. What is the magnitude of the magnetic induction at point P outside of the cable at a distance r from the axis?

A. Zero

B.

C.

D.

E.

(GR9277 #09)

Solution:

The inner and outer conductors carry equal and opposite currents. Thus, the magnitude of the magnetic induction outside the coaxial cable is zero.

Answer: A 

Electromagnetism - Drift Velocity

A wire of diameter 0.02 meter contains 10²⁸ free electrons per cubic meter. For an electric current of 100 amperes, the drift velocity for free electrons in the wire is most nearly

A. 0.6 × 10⁻²⁹ m/s
B. 1 × 10⁻¹⁹ m/s
C. 5 × 10⁻¹⁰ m/s
D. 2 × 10⁻⁴ m/s
E. 8 × 10³ m/s

(GR8677 #09)

Solution:

Drift velocity is the average velocity of a carrier that is moving under the influence of an electric field.

Velocity: v =  L/t

In a wire with length L and cross sectional area A, there are n electrons with charge q_e per cubic meter.

Total number of mobile electrons in the wire, Q = nq_eLA

Current: I = Q/t = nq_eLA/t = nq_evA
v = I/nq_eA

q_e = charge of an electron = 1.6 × 10⁻¹⁹ C
n = 10²⁸ electrons/m³
A = πr² =  0.5π × 10⁻⁴
I =100 Ampere

v = 10² / (10²⁸× 1.6 × 10⁻¹⁹× 0.5π × 10⁻⁴)
= 10^2−28+19+4 / (1.6 × 0.5π)
≈ 10⁻⁴

Answer: D

Electromagnetism - Gauss' Law

Five positive charges of magnitude q are arranged symmetrically around the circumference of a circle of radius r. What is the magnitude of the electric field at the center of the circle? (k = 1/4πε₀)

A. 0
B. kq/r²
C. 5kq/r²
D. (kq/r²) cos (2π/5)
E. (5kq/r²) cos (2π/5)

(GR0177 #09)

Solution:

Gauss’ Law:  

There is no q_enc inside the Gaussian surface → E = 0  

Answer: A