Two resistors of resistance `R_(1)` and `R_(2)` having `R_(1) gt R_(2)` are connected in parallel. For equivalent resistance R, the correct statement is

Options

A. `R gt r_(1) + r_(2)`
B. `R gt sqrt(r_(1) r_(2))`
C. `r_(1) lt R lt r_(2)`
D. `R lt r_(1)`

Answer

To solve the problem of finding the equivalent resistance \( R \) of two resistors \( R_1 \) and \( R_2 \) connected in parallel, we can follow these steps:

### Step 1: Understand the Parallel Connection
In a parallel connection, the voltage across each resistor is the same, and the total current is the sum of the currents through each resistor.

### Step 2: Use the Formula for Equivalent Resistance
The formula for the equivalent resistance \( R \) of two resistors \( R_1 \) and \( R_2 \) in parallel is given by:
\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 3: Rearrange the Formula
To find \( R \), we can rearrange the formula:
\[ \frac{1}{R} = \frac{R_2 + R_1}{R_1 R_2} \]
Thus, we can express \( R \) as:
\[ R = \frac{R_1 R_2}{R_1 + R_2} \]

### Step 4: Analyze the Result
Since \( R_1 > R_2 \), it follows that:
- \( R_1 + R_2 > R_1 \)
- \( R_1 + R_2 > R_2 \)

This implies that:
\[ R < R_1 \quad \text{and} \quad R < R_2 \]
Therefore, the equivalent resistance \( R \) is less than either \( R_1 \) or \( R_2 \).

### Step 5: Conclusion
The correct statement regarding the equivalent resistance \( R \) is that it is less than both \( R_1 \) and \( R_2 \).