Two resistors of resistance `R_1` and `R_2` having `R_1>R_2` are connected in parallel. For equivalent resistance R, the correct statement is

Options

A. `RgtR_1+R_2`
B. `R_1ltRltR_2`
C. `R_2ltRlt(R_1+R_2)`
D. `RltR_1`

Answer

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

### Step 1: Understand the Configuration
We have two resistors, \( R_1 \) and \( R_2 \), connected in parallel. We know that \( R_1 > R_2 \).

### Step 2: Write the Formula for Equivalent Resistance
The formula for the equivalent resistance \( R \) of two resistors in parallel is given by:

\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 3: Substitute Values
For better understanding, let's assume some values for \( R_1 \) and \( R_2 \). Let \( R_1 = 5 \, \Omega \) and \( R_2 = 3 \, \Omega \) (since \( R_1 > R_2 \)).

### Step 4: Calculate the Equivalent Resistance
Substituting the assumed values into the formula:

\[ \frac{1}{R} = \frac{1}{5} + \frac{1}{3} \]

Finding a common denominator (which is 15):

\[ \frac{1}{R} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15} \]

Now, taking the reciprocal to find \( R \):

\[ R = \frac{15}{8} = 1.875 \, \Omega \]

### Step 5: Analyze the Statements
Now we need to analyze the statements provided in the question:

1. **Statement A**: \( R > R_1 + R_2 \)
- \( R_1 + R_2 = 5 + 3 = 8 \)
- \( 1.875 \) is not greater than \( 8 \). This statement is false.

2. **Statement B**: \( R_2 > R \)
- \( R_2 = 3 \) and \( R = 1.875 \)
- \( 3 > 1.875 \). This statement is true.

3. **Statement C**: \( R_1 + R_2 > R \)
- We already calculated \( R_1 + R_2 = 8 \) and \( R = 1.875 \)
- \( 8 > 1.875 \). This statement is true.

4. **Statement D**: \( R < R_1 \)
- \( R_1 = 5 \) and \( R = 1.875 \)
- \( 1.875 < 5 \). This statement is true.

### Step 6: Conclusion
The correct statements are B, C, and D. However, if we are to select the most relevant correct statement regarding the equivalent resistance \( R \), we can conclude that:

**Final Answer**: \( R < R_1 \) (Statement D is correct).

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