A battery delivers equal power individually across `4 Omega ` and `16 Omega ` . Internal resistance of the cell is

Options

A. `8 Omega`
B. ` 6 Omega `
C. ` 12 Omega `
D. ` 20 Omega`

Answer

To solve the problem step by step, we need to find the internal resistance of the battery given that it delivers equal power across two resistors of 4 ohms and 16 ohms.

### Step 1: Define the power equations for both resistors.
The power delivered across a resistor can be calculated using the formula:
\[ P = I^2 R \]
where \( I \) is the current through the resistor and \( R \) is the resistance.

For the 4-ohm resistor:
\[ P_1 = I_1^2 \cdot 4 \]

For the 16-ohm resistor:
\[ P_2 = I_2^2 \cdot 16 \]

### Step 2: Calculate the currents \( I_1 \) and \( I_2 \).
Using Ohm's law, the current through each resistor can be calculated as follows:

For the 4-ohm resistor:
\[ I_1 = \frac{E}{4 + r} \]

For the 16-ohm resistor:
\[ I_2 = \frac{E}{16 + r} \]

where \( E \) is the EMF of the battery and \( r \) is the internal resistance.

### Step 3: Substitute the currents into the power equations.
Now we substitute \( I_1 \) and \( I_2 \) into the power equations:

For the 4-ohm resistor:
\[ P_1 = \left(\frac{E}{4 + r}\right)^2 \cdot 4 \]

For the 16-ohm resistor:
\[ P_2 = \left(\frac{E}{16 + r}\right)^2 \cdot 16 \]

### Step 4: Set the powers equal to each other.
Since the problem states that the power delivered is equal for both resistors, we can set the two power equations equal to each other:
\[ \left(\frac{E}{4 + r}\right)^2 \cdot 4 = \left(\frac{E}{16 + r}\right)^2 \cdot 16 \]

### Step 5: Simplify the equation.
We can cancel \( E^2 \) from both sides (assuming \( E \neq 0 \)):
\[ \frac{4}{(4 + r)^2} = \frac{16}{(16 + r)^2} \]

Cross-multiplying gives:
\[ 4(16 + r)^2 = 16(4 + r)^2 \]

### Step 6: Expand both sides.
Expanding both sides:
\[ 4(256 + 32r + r^2) = 16(16 + 8r + r^2) \]
\[ 1024 + 128r + 4r^2 = 256 + 128r + 16r^2 \]

### Step 7: Rearrange the equation.
Rearranging gives:
\[ 1024 - 256 = 16r^2 - 4r^2 \]
\[ 768 = 12r^2 \]

### Step 8: Solve for \( r \).
Dividing both sides by 12:
\[ r^2 = \frac{768}{12} = 64 \]
Taking the square root:
\[ r = 8 \, \Omega \]

### Conclusion:
The internal resistance of the cell is \( r = 8 \, \Omega \).

---