Two resistor of resistance `R_(1)=(6+-0.09)Omega` and `R_(2)=(3+-0.09)Omega` are connected in parallel the equivalent resistance R with error (in `Omega)`

Options

A. `R=(2+-0.04)`
B. `R=(2+-0.05)`
C. `R=(9+-0.18)`
D. `R=(2+-0.18)`

Explanation

To find the equivalent resistance \( R \) of two resistors \( R_1 \) and \( R_2 \) connected in parallel, along with the associated error, we can follow these steps:

### Step 1: Write down the formula for equivalent resistance in parallel
The formula for the equivalent resistance \( R \) when two resistors are connected in parallel is given by:
\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]
This can also be expressed as:
\[ R = \frac{R_1 \cdot R_2}{R_1 + R_2} \]

### Step 2: Substitute the values of \( R_1 \) and \( R_2 \)
Given:
- \( R_1 = 6 \, \Omega \) with an uncertainty of \( \Delta R_1 = 0.09 \, \Omega \)
- \( R_2 = 3 \, \Omega \) with an uncertainty of \( \Delta R_2 = 0.09 \, \Omega \)

Substituting these values into the formula:
\[ R = \frac{6 \cdot 3}{6 + 3} = \frac{18}{9} = 2 \, \Omega \]

### Step 3: Calculate the uncertainty in the equivalent resistance
To find the uncertainty in the equivalent resistance, we will use the formula for the propagation of uncertainty. The differential form gives:
\[ dR = R^2 \left( \frac{dR_1}{R_1^2} + \frac{dR_2}{R_2^2} \right) \]
Where:
- \( dR_1 = 0.09 \, \Omega \)
- \( dR_2 = 0.09 \, \Omega \)

### Step 4: Substitute the values into the uncertainty formula
Now substituting the values:
\[ dR = R^2 \left( \frac{0.09}{6^2} + \frac{0.09}{3^2} \right) \]
Calculating each term:
- \( 6^2 = 36 \)
- \( 3^2 = 9 \)

Thus:
\[ dR = (2^2) \left( \frac{0.09}{36} + \frac{0.09}{9} \right) \]
\[ dR = 4 \left( \frac{0.09}{36} + \frac{0.09}{9} \right) \]
Calculating the fractions:
\[ \frac{0.09}{36} = 0.0025 \quad \text{and} \quad \frac{0.09}{9} = 0.01 \]
Adding these:
\[ 0.0025 + 0.01 = 0.0125 \]
Now substituting back:
\[ dR = 4 \cdot 0.0125 = 0.05 \, \Omega \]

### Step 5: Write the final result
The equivalent resistance with its uncertainty is:
\[ R = 2 \pm 0.05 \, \Omega \]