PHYS 230/281 Midterm Exam Study Guidance

Q5. For two spheres with charges Q1 and Q2 and identical radii R, nearly touching, what must be the ratio Q2/Q1 for the total electric potential at point P to be zero? (Assume the potential is zero at infinity.)

Background

Topic: Electric Potential due to Point Charges

This question tests your understanding of how to calculate the electric potential at a point due to multiple point charges, and how to set up equations for the condition that the net potential at a specific point is zero.

Key Terms and Formulas

• Electric Potential (V): The potential at a distance r from a point charge Q is given by:

  $V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$

• Superposition Principle: The total potential at a point due to multiple charges is the algebraic sum of the potentials due to each charge.

Step-by-Step Guidance

1. Write the expression for the electric potential at point P due to charge Q1:

   $V_1 = \frac{1}{4\pi\varepsilon_0} \frac{Q_1}{r_1}$

2. Write the expression for the electric potential at point P due to charge Q2:

   $V_2 = \frac{1}{4\pi\varepsilon_0} \frac{Q_2}{r_2}$

3. Apply the superposition principle to find the total potential at point P:

   $V_{\text{total}} = V_1 + V_2$

4. Set the total potential at point P to zero (as required by the problem):

   $V_1 + V_2 = 0$

5. Substitute the expressions for $V_1$ and $V_2$ and rearrange to solve for the ratio $Q_2/Q_1$.

Try solving on your own before revealing the answer!

Final Answer: $Q_2/Q_1 = -r_2/r_1$

By setting $\frac{Q_1}{r_1} + \frac{Q_2}{r_2} = 0$, you solve for $Q_2/Q_1$ as $-r_2/r_1$. The negative sign indicates the charges must be of opposite sign for the potentials to cancel at P.

Q9. In the circuit shown, the EMF source E has negligible internal resistance, and C₁ ≠ C₂. Which statement is correct?

Background

Topic: Capacitors in Parallel

This question tests your understanding of how capacitors behave when connected in parallel, including charge, voltage, and energy relationships.

Key Terms and Formulas

• Capacitors in Parallel: The voltage across each capacitor is the same and equal to the EMF of the source.

• Total Capacitance: $C_{\text{eq}} = C_1 + C_2$

• Charge on Each Capacitor: $Q_i = C_i V$

• Energy Stored: $U = \frac{1}{2} C V^2$

Step-by-Step Guidance

1. Recall that in a parallel circuit, the voltage across each capacitor is equal to the EMF, $V = \mathcal{E}$.

2. Determine the charge on each capacitor using $Q_i = C_i V$.

3. Compare the charges and energies stored in each capacitor, noting that $C_1 \neq C_2$.

4. Consider the total current and whether the EMF source dissipates energy (an ideal source does not dissipate energy).

Try solving on your own before revealing the answer!

Final Answer: The capacitors have equal potential difference.

In a parallel connection, each capacitor is directly connected across the EMF source, so $V_1 = V_2 = \mathcal{E}$ regardless of their capacitances.