Electromagnetism - Capacitor

Two real capacitors of equal capacitance (C₁ = C₂) are shown in the figure. Initially, while the switch S is open, one of the capacitors is uncharged and the other carries charge Q₀. The energy stored in the charged capacitor is U₀. Sometime after the switch is closed, the capacitors C₁ and C₂ carry charges Q₁ and Q₂, respectively; the voltages across the capacitors are V₁ and V₂; and the energies stored in the capacitors are U₁ and U₂.Which of the following statements is INCORRECT?

A. Q₀ = ½ (Q₁ + Q₂)
B. Q₁ = Q₂
C. V₁ = V₂
D. U₁ = U₂
E. U₀ = U₁ + U₂

(GR9677 #25)

Solution:

Initial charge: Q₀
Charge is conserved and distribute evenly
Q₀ = ½ (Q₁ + Q₂)
(A) CORRECT

C = Q/V
For parallel capacitor V = V₁ = V₂
Given C₁ = C₂ → Q₁ = Q₂
(B) and (C) CORRECT

U = ½QV
V₁ = V₂ and Q₁ = Q₂ 
→ U₁ = U₂
(D) CORRECT

U = ½QV → Q = 2U/V
From (A) Q₀ = ½ (Q₁ + Q₂),
2U₀/V₀ = ½ (2U₁/V₁ + 2U₂/V₂)
V₀ = V₁ = V₂ → 2U₀ = U₁ + U₂

(E) INCORRECT

Answer: E

Nuclear & Particle Physics - Cosmic Rays

In experiments located deep underground the two types of cosmic rays that most commonly reach the experimental apparatus are

A. alpha particles and neutrons
B. protons and electrons
C. iron nuclei and carbon nuclei
D. muons and neutrinos
E. positrons and electrons

(GR9277 #25)

Solution:

Since the experiment is located deep underground, the particles must be essentially massless and able to pass through or do not interact with matter easily → muons and neutrinos.

Muon:

• "heavy" electron (200 times the mass of the electrons)
• highly penetrating

Neutrinos:

• almost no mass
• travel close to the speed of light
• weakly interact with other

Answer: D



Notes:

• Alpha particles and neutrons have high kinetic energy but very short penetrating depth because they are quite massive, can't go through the Earth.
• Protons and electrons don’t penetrate much as they interact easily with matter. Protons are also pretty massive.
• Iron and carbon nuclei are very heavy and interact very easily with matter.
• Positrons and electrons have the same mass and are highly interacting with matter. Positron also has a very short life span.

Electromagnetism - Electric and Magnetic Forces

A charge particle is released from rest in a region where there is a constant electric field and a constant magnetic field. If the two fields are parallel to each other, the path of the particle is a

A. circle
B. parabola
C. helix
D. cycloid
E. straight line

(GR8677 #25)

Solution:



Electric Force:
→ the path of the particle is parallel to the electric field.
→ the velocity of particle has the same direction with the electric field.

Magnetic Force/Lorentz Force:  

If
→ no magnetic field contribution
→ the path of the particle is a straight line, parallel to the electric field.

Answer: E

Classical Mechanics - Rotational Motion

Seven pennies are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure. Each penny is a uniform disk of mass m and radius r. What is the moment of inertia of the system of seven pennies about an axis that passes through the center of the central penny and is normal to the plane of the pennies?

A. (7/2) mr²
B. (13/2) mr²
C. (29/2) mr²
D. (49/2) mr²
E. (55/2) mr²

(GR0177 #25)

Solution:

Parallel axis theorem: I = ml² + I_CM

Moment inertia of each penny (a uniform disk): I = ½ mr²

with l = 2r,

I = m(2r)² + ½ mr² = (9/2) mr²

I_total =  I_{6 outer pennies} + I_{1 central penny}

I_total = 6 × (9/2) mr²  + ½ mr² = (55/2) mr²

Answer: E