When two resistors of `(600pm3)` ohm and `(300pm 6)` ohm are connected in parallel, value of the equivalent resistance is

Options

A. `200 ohm pm 1.5 %`
B. 200 ohm `pm 3.5%`
C. 200 ohm `pm 9%`
D. 200 ohm

Answer

To find the equivalent resistance of two resistors connected in parallel and the associated error, we can follow these steps:

### Step 1: Identify the given values
We have two resistors:
- \( R_1 = 600 \, \Omega \) with an uncertainty of \( \pm 3 \, \Omega \)
- \( R_2 = 300 \, \Omega \) with an uncertainty of \( \pm 6 \, \Omega \)

### Step 2: Use the formula for equivalent resistance in parallel
The formula for the equivalent resistance \( R_{eq} \) of two resistors in parallel is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 3: Substitute the values into the formula
Substituting the values of \( R_1 \) and \( R_2 \):

\[ \frac{1}{R_{eq}} = \frac{1}{600} + \frac{1}{300} \]

### Step 4: Calculate the right-hand side
To calculate the right-hand side, we find a common denominator:

\[ \frac{1}{600} + \frac{1}{300} = \frac{1}{600} + \frac{2}{600} = \frac{3}{600} = \frac{1}{200} \]

### Step 5: Find the equivalent resistance
Now, taking the reciprocal gives us:

\[ R_{eq} = 200 \, \Omega \]

### Step 6: Calculate the uncertainty in the equivalent resistance
To find the uncertainty in \( R_{eq} \), we differentiate the formula for \( R_{eq} \):

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

Differentiating both sides gives:

\[ -\frac{dR_{eq}}{R_{eq}^2} = -\frac{dR_1}{R_1^2} - \frac{dR_2}{R_2^2} \]

Rearranging this gives:

\[ dR_{eq} = R_{eq}^2 \left( \frac{dR_1}{R_1^2} + \frac{dR_2}{R_2^2} \right) \]

### Step 7: Substitute the uncertainties
Substituting the values:

- \( dR_1 = 3 \, \Omega \)
- \( dR_2 = 6 \, \Omega \)

We have:

\[ dR_{eq} = (200)^2 \left( \frac{3}{600^2} + \frac{6}{300^2} \right) \]

Calculating each term:

\[ = 40000 \left( \frac{3}{360000} + \frac{6}{90000} \right) \]

Simplifying:

\[ = 40000 \left( \frac{3}{360000} + \frac{24}{360000} \right) = 40000 \left( \frac{27}{360000} \right) \]

Calculating:

\[ = 40000 \times \frac{27}{360000} = \frac{1080000}{360000} = 3 \, \Omega \]

### Step 8: Calculate the percentage error
The percentage error is given by:

\[ \text{Percentage error} = \left( \frac{dR_{eq}}{R_{eq}} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage error} = \left( \frac{3}{200} \right) \times 100 = 1.5\% \]

### Final Result
Thus, the equivalent resistance is:

\[ R_{eq} = 200 \, \Omega \pm 1.5\% \]