Capacitor Charging Time & Electrostatic Equilibrium Problems – PHYS 202 Exam Prep

Q1. What is the correct ranking of the circuits based on the amount of time for the capacitor to charge completely, from the shortest to the longest amount of time?

Background

Topic: RC Circuits & Time Constant

This question tests your understanding of how the arrangement of resistors in a circuit affects the charging time of a capacitor. The time it takes for a capacitor to charge depends on the circuit's resistance and capacitance.

Key Terms and Formulas

• Equivalent Resistance ($R_{eq}$): The total resistance seen by the capacitor in each circuit.

• Time Constant ($\tau$): $\tau = R_{eq}C$

• Capacitance ($C$): The ability of a capacitor to store charge.

Step-by-Step Guidance

1. Analyze each circuit diagram to determine how the resistors are arranged (series or parallel).

2. Calculate the equivalent resistance for each circuit. For example, resistors in series add, while resistors in parallel combine as $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$.

3. Use the formula $\tau = R_{eq}C$ to find the time constant for each circuit.

4. Rank the circuits based on their time constants: the smaller the time constant, the faster the capacitor charges.

Try solving on your own before revealing the answer!

Q2. The figure shows two tiny 5.0-g spheres suspended from very light 1.0-m-long threads. The spheres repel each other after each one is given the same positive charge and hang at rest when $\theta = 4.1^\circ$. What is the charge on each sphere?

Background

Topic: Electrostatics & Equilibrium

This question tests your ability to apply Newton's Laws and Coulomb's Law to a system in equilibrium, where two charged spheres are suspended and repel each other.

Key Terms and Formulas

• Coulomb's Law: $F = k \frac{q^2}{r^2}$

• Newton's Second Law: Forces in equilibrium ($\sum F_x = 0$, $\sum F_y = 0$)

• Trigonometric relationships: $\tan \theta = \frac{F_{elec}}{mg}$

• Mass ($m$), gravitational acceleration ($g$), thread length ($L$), and angle ($\theta$)

Step-by-Step Guidance

1. Draw a free-body diagram for one sphere, showing tension, gravity, and electrostatic force.

2. Write equations for equilibrium in the $x$ and $y$ directions: $T \sin \theta = F_{elec}$ and $T \cos \theta = mg$.

3. Divide the $x$ equation by the $y$ equation to eliminate $T$: $\tan \theta = \frac{F_{elec}}{mg}$.

4. Express $F_{elec}$ using Coulomb's Law: $F_{elec} = k \frac{q^2}{r^2}$, where $r$ is the distance between the spheres.

5. Set up the equation for $q$ in terms of known quantities: $q = \sqrt{\frac{mg r^2 \tan \theta}{k}}$.

Try solving on your own before revealing the answer!

Final Answer: (Sample for Q2) 22 nC

$q = \sqrt{\frac{mg r^2 \tan \theta}{k}}$

Plugging in the values for mass, gravity, distance, angle, and Coulomb's constant gives the charge on each sphere.