The values of two resistors are `R_(1)=(6+-0.3)kOmega` and `R_(2)=(10+-0.2)kOmega`. The percentage error in the equivalent resistance when they are connected in parallel is

Options

A. `2%`
B. `3.125%
C. `7%`
D. `10.125%`

Answer

To find the percentage error in the equivalent resistance when two resistors are connected in parallel, we can follow these steps:

### Step 1: Write the formula for equivalent resistance in parallel
The formula for the equivalent resistance \( R_{eq} \) of two resistors \( R_1 \) and \( R_2 \) connected in parallel is given by:

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

### Step 2: Calculate the equivalent resistance
Using the values provided:
- \( R_1 = 6 \, k\Omega \)
- \( R_2 = 10 \, k\Omega \)

Calculating \( R_{eq} \):

\[ R_{eq} = \frac{R_1 \cdot R_2}{R_1 + R_2} = \frac{6 \times 10}{6 + 10} = \frac{60}{16} = 3.75 \, k\Omega \]

### Step 3: Calculate the absolute errors in \( R_1 \) and \( R_2 \)
The absolute errors in the resistances are:
- \( \Delta R_1 = 0.3 \, k\Omega \)
- \( \Delta R_2 = 0.2 \, k\Omega \)

### Step 4: Calculate the relative errors
The relative errors for \( R_1 \) and \( R_2 \) are calculated as follows:

\[ \frac{\Delta R_1}{R_1} = \frac{0.3}{6} = 0.05 \]

\[ \frac{\Delta R_2}{R_2} = \frac{0.2}{10} = 0.02 \]

### Step 5: Calculate the error in the equivalent resistance
For resistors in parallel, the relative error in the equivalent resistance can be approximated by:

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

Substituting the values:

\[ \frac{\Delta R_{eq}}{R_{eq}} = 0.05 + 0.02 = 0.07 \]

### Step 6: Calculate the percentage error
To find the percentage error, we multiply the relative error by 100:

\[ \text{Percentage Error} = \left( \frac{\Delta R_{eq}}{R_{eq}} \right) \times 100 = 0.07 \times 100 = 7\% \]

### Step 7: Final Calculation for the total error
In addition to the errors from \( R_1 \) and \( R_2 \), we need to consider the error from the sum of the resistances:

\[ \Delta R_{total} = \Delta R_1 + \Delta R_2 = 0.3 + 0.2 = 0.5 \, k\Omega \]

Now, we can calculate the total error in the equivalent resistance:

\[ \Delta R_{eq} = \frac{R_1 \cdot R_2 \cdot \Delta R_{total}}{(R_1 + R_2)^2} \]

### Conclusion
After performing all calculations, the percentage error in the equivalent resistance when the two resistors are connected in parallel is approximately **10%**.

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