An element with `emf epsilon ` and interval resistance `r` is connected across an external resistance `R`. The maximum power in external circuit is `9 W`. The current flowing through the circuit in these conditions is `3 A`. Then which of the following is // are correct ?

Options

A. ` epsilon = 6 V`
B. ` r = R `
C. `r = 1 Omega`
D. `r = 3 Omega`

Explanation

To solve the problem step by step, we will analyze the given information and apply the relevant formulas.

### Step 1: Understand the Circuit
We have an EMF source (ε), an internal resistance (r), and an external resistance (R) connected in series. The current flowing through the circuit is given as 3 A, and the maximum power in the external circuit is given as 9 W.

### Step 2: Use the Power Formula
The power (P) dissipated across the external resistance R can be expressed as:
\[ P = I^2 R \]
Where:
- \( I \) is the current flowing through the circuit.

Given that the maximum power \( P_{max} = 9 \, W \) and \( I = 3 \, A \), we can rearrange the formula to find R:
\[ 9 = (3)^2 R \]
\[ 9 = 9R \]
\[ R = 1 \, \Omega \]

### Step 3: Use the Maximum Power Transfer Theorem
For maximum power transfer, the external resistance (R) should be equal to the internal resistance (r):
\[ R = r \]
Thus, we have:
\[ r = 1 \, \Omega \]

### Step 4: Find the EMF (ε)
Using Ohm's law, we can find the EMF (ε) of the source. The total resistance in the circuit is:
\[ R + r = 1 + 1 = 2 \, \Omega \]
Using Ohm's law:
\[ I = \frac{E}{R + r} \]
Substituting the known values:
\[ 3 = \frac{E}{2} \]
Thus:
\[ E = 3 \times 2 = 6 \, V \]

### Step 5: Summary of Results
From the calculations, we find:
- EMF (ε) = 6 V
- Internal Resistance (r) = 1 Ω
- External Resistance (R) = 1 Ω

### Conclusion
The correct answers based on the calculations are:
- EMF (ε) = 6 V
- Internal Resistance (r) = 1 Ω
- External Resistance (R) = 1 Ω