Two condensers of 20 and 30 microfarads are connected in series across a 200volt D.C. supply. Find the charge on each condenser ?

Options

A. `2400muC`
B. `4200muC`
C. `2600muC`
D. `3000muC`

Answer

To solve the problem of finding the charge on each condenser when two capacitors of 20 microfarads and 30 microfarads are connected in series across a 200-volt DC supply, we can follow these steps:

### Step 1: Identify the capacitance values
Let:
- \( C_1 = 20 \, \mu F \)
- \( C_2 = 30 \, \mu F \)

### Step 2: Calculate the equivalent capacitance for capacitors in series
The formula for the equivalent capacitance \( C_{eq} \) for capacitors in series is given by:
\[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \]
Substituting the values:
\[ \frac{1}{C_{eq}} = \frac{1}{20 \, \mu F} + \frac{1}{30 \, \mu F} \]

### Step 3: Find a common denominator and simplify
The least common multiple (LCM) of 20 and 30 is 60. Therefore, we can rewrite the equation:
\[ \frac{1}{C_{eq}} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \]
Now, taking the reciprocal to find \( C_{eq} \):
\[ C_{eq} = \frac{60}{5} = 12 \, \mu F \]

### Step 4: Calculate the total charge using the equivalent capacitance
The charge \( Q \) stored in the equivalent capacitor can be calculated using the formula:
\[ Q = C_{eq} \times V \]
Where \( V = 200 \, V \):
\[ Q = 12 \, \mu F \times 200 \, V = 2400 \, \mu C \]

### Step 5: Determine the charge on each condenser
In a series circuit, the charge on each capacitor is the same. Therefore:
\[ Q_1 = Q_2 = Q = 2400 \, \mu C \]

### Final Answer
The charge on each condenser is:
- Charge on the 20 microfarad capacitor \( C_1 = 2400 \, \mu C \)
- Charge on the 30 microfarad capacitor \( C_2 = 2400 \, \mu C \)

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