Nuclear & Particle Physics - Helium

If a singly ionized Helium atom in an n = 4 state emits a photon of wavelength 470 nanometers, which of the following gives the approximate final energy level E_f  of the atom, and the n value, of n_f  this final state?

	Ef  (eV)	nf
A.	− 6.0	3
B.	− 6.0	2
C.	− 14	2
D.	− 14	1
E.	− 52	1

(GR9677 #40)

Solution:

E_photon = E_i  − E_f 

Helium: 2 electrons, 2 protons, 2 neutron
Singly ionized Helium, He⁺:1 electrons, 2 protons, 2 neutron
He⁺ → Hydrogen-like atom

Bohr's Equation for Hydrogen-like atom: E_n =  −13.6 Z²/n² eV

For Helium, Z = 2,
E_i = E_(n =4)  = − 13.6 (2)² /4² = − 13.6 /4 ≈ − 3.4 eV

E_photon  = hν = hc / λ 
with
h = 6.63 × 10⁻³⁴ Joule.second = 4.1 × 10⁻¹⁵ eV.second
c = 3 × 10⁸ m/s
λ  = 470 nm = 470 × 10⁻⁹ m = 4.7 × 10⁻⁷ m

E_photon = (4.1 × 10⁻¹⁵)(3 × 10⁸) / (4.7 × 10⁻⁷) ≈ 3 eV

E_f  = E_i − E_photon  
=  − 3.4 − 3
= − 6.4  eV

To find n_f :

n_f ² = −13.6 Z²/ E_f 
= −13.6 (2)²/ (− 6.4)
= 54.4/6.4 ≈ 9
n_f  = 3

Answer: A

Classical Mechanics - Circular Motion

A rigid cylinder rolls at constant speed without slipping on top of a horizontal plane surface. The acceleration of a point on the circumference of the cylinder at the moment when the point touches the plane is

A. directed forward
B. directed backward
C. directed up
D. directed down
E. zero

(GR9277 #40)

Solution:

Constant speed → no tangential acceleration → Uniform Circular Motion



When the point touches the plane,

 is directed up

Answer: C

Nuclear & Particle Physics - Radioactive

An experimenter measures 9924 counts during one hour from a radioactive sample. From this number the counting rate of the sample can be estimated with a standard deviation of most nearly 

A. 100
B. 200
C. 300
D. 400
E. 500

(GR8677 #40)

Solution:

Radioactive decay can be described by Poisson Distribution.

Poisson Distribution (PD):
Probability distribution of discrete events over an interval (time. distance, etc)

In PD, Standard Deviation, σ = √μ (see Notes)
μ = expected number of events

Given: μ = 9924 ≈ 10000
σ = √10000 = 100

Answer: A


Notes: (click to enlarge)

Electromagnetism - RL Circuit

 

In the circuit shown above, the switch S is closed at t = 0. Which of the following best represents the voltage across the inductor, as seen on an oscilloscope?

(GR0177 #40)

Solution:

The voltage drops across the resistor and the inductor → (A), (B), (C) are FALSE.

The time constant when the voltage decreases slowly is



Answer: D