Consider two conductors. One of them has a capacity of 2 units and the capacity of the other is unknown. They are charged until their potential are 4 and 5 units respectively. The two conductors are now connected by a wire when their common potential is found to be 4.6 units. Then the unknown capacity has the value (in the same units as above)

Options

A. 6
B. 5
C. 4
D. 3

Answer

To solve the problem, we need to apply the principle of conservation of charge. Here's a step-by-step solution:

### Step 1: Define the known values
Let:
- \( C_1 = 2 \) units (the capacitance of the first conductor)
- \( V_1 = 4 \) units (the potential of the first conductor)
- \( V_2 = 5 \) units (the potential of the second conductor)
- \( C_2 = x \) units (the capacitance of the second conductor, which we need to find)
- The common potential after connecting the conductors is \( V_f = 4.6 \) units.

### Step 2: Calculate the initial charges on each conductor
The charge on a capacitor is given by the formula:
\[ Q = C \times V \]

Thus, the initial charges on the conductors are:
- Charge on the first conductor:
\[ Q_1 = C_1 \times V_1 = 2 \times 4 = 8 \text{ units} \]
- Charge on the second conductor:
\[ Q_2 = C_2 \times V_2 = x \times 5 \]

### Step 3: Write the conservation of charge equation
According to the conservation of charge, the total initial charge before connecting the conductors is equal to the total charge after they are connected:
\[ Q_{initial} = Q_{final} \]
Thus, we can write:
\[ Q_1 + Q_2 = (C_1 + C_2) \times V_f \]
Substituting the values we have:
\[ 8 + 5x = (2 + x) \times 4.6 \]

### Step 4: Expand and simplify the equation
Expanding the right-hand side:
\[ 8 + 5x = 2 \times 4.6 + x \times 4.6 \]
\[ 8 + 5x = 9.2 + 4.6x \]

### Step 5: Rearranging the equation
Now, we will rearrange the equation to isolate \( x \):
\[ 8 + 5x - 4.6x = 9.2 \]
\[ 8 + 0.4x = 9.2 \]
Subtracting 8 from both sides:
\[ 0.4x = 1.2 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 0.4:
\[ x = \frac{1.2}{0.4} = 3 \text{ units} \]

### Conclusion
The unknown capacitance \( C_2 \) is \( 3 \) units.

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