Two capacitors C1 and C2 are charged to 120 V and 200 V, respectively. It is found that by connectin

Q: Why did we assume they were connected with opposite polarities?

If they were connected with the same polarity (positive to positive), their charges would add up ($q_1 + q_2$), and the net charge would be strictly positive. A positive net charge divided by a positive total capacitance can never yield a zero potential. Thus, opposite polarity is the only physical way to achieve zero volts.

Q: What happens to the energy stored in the capacitors?

Since the final potential is zero, the final energy of the system is also zero ($\frac{1}{2} C_{\text{eq}} V^2 = 0$). All the initial electrostatic potential energy is completely dissipated as heat and electromagnetic radiation in the connecting wires during the charge redistribution.

Q: Can the potential be zero if $C_1 = C_2$?

Only if their initial voltages were also equal ($V_1 = V_2$). In this specific problem, since $V_1 \neq V_2$, the capacitances must be unequal to ensure the charges ($CV$) balance out.

Q: Is the charge on each plate zero finally?

Yes. When connected with opposite polarities, the $+q$ from one plate perfectly neutralizes the $-q$ from the other plate, leaving zero net charge on all plates.

Q: What if the capacitors were connected in series instead?

The phrase 'connecting them together' to find a 'potential on each one' implies a closed loop where they share charge until potentials equalize, which is effectively a parallel connection. In a pure series circuit without a closed loop, charge doesn't flow to equalize potentials in this manner.

Question

Two capacitors $C_1$ and $C_2$ are charged to $120	ext{ V}$ and $200	ext{ V}$, respectively. It is found that by connecting them together, the potential on each one can be made zero. Then,

Options:

(A) $5C_1 = 3C_2$

(B) $3C_1 = 5C_2$

(C) $3C_1 + 5C_2 = 0$

(D) $9C_1 = 4C_2$

Sigma Percentile · Accepted Answer

### The Mystery of the Zero Potential

Imagine you have two fully charged capacitors. One is buzzing with $120	ext{ V}$ and the other is packed with $200	ext{ V}$. You connect them together, and suddenly... nothing. The potential across both of them drops to absolute zero. How is this physically possible? Let's unravel this mystery.

### Analyzing the Setup

When two charged capacitors are connected in parallel, they share their charge until they reach a **common potential**. The formula for this common potential is the total net charge divided by the total capacitance:

$$V_{	ext{common}} = \frac{Q_{	ext{net}}}{C_1 + C_2}$$

Now, here is the catch. The problem explicitly states that this common potential becomes zero. Since the denominator $(C_1 + C_2)$ cannot be zero (capacitance is always positive), the numerator must be zero!

$$Q_{	ext{net}} = 0$$

### The Master Equation

How can the net charge be zero? If we connect the positive plate of $C_1$ to the positive plate of $C_2$, the charges would add up $(q_1 + q_2)$, and the potential would never be zero. Therefore, we must have connected them with **opposite polarities**—the positive plate of one to the negative plate of the other. In this case, the net charge is the difference between their magnitudes:

$$Q_{	ext{net}} = q_1 - q_2 = 0 \implies q_1 = q_2$$

This tells us that for the final potential to be zero, the initial charges on both capacitors must have been exactly equal in magnitude.

### Final Calculation

Let's express these charges in terms of capacitance and voltage. We know that $q = CV$.

For the first capacitor:
$$q_1 = C_1 	imes 120$$

For the second capacitor:
$$q_2 = C_2 	imes 200$$

Equating them as per our condition:

$$120 C_1 = 200 C_2$$

To simplify, we can divide both sides by their greatest common divisor, which is $40$:

$$3 C_1 = 5 C_2$$

And there we have it! The elegant relationship between the two capacitances that allows them to completely neutralize each other. Always remember to check the polarity when connecting capacitors!

Animated Solution for Physics - Electrostatics: Two capacitors $C_{1}$ and $C_{2}$ are charged to $120 V$ and $200 V$ , respectively. It is found that by connecting them together, the potential on each one can be made zero. Then,

Select Answer:

Visualized Solution

Initial State

Condition for Zero Potential

Initial Charges

Equating Charges

Final Ratio

The Way Forward

The Sigma Insight: Combination of Capacitors

The Mystery of the Zero Potential

Analyzing the Setup

The Master Equation

Final Calculation

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