The possible percentage error in computing the parallel resistance R of three resistances `1/R=(1)/R_1+ 1/R_2 1/R_3` from the formula , if `R_1, R_2, R_3` are each in error by plus 1.2% is :

Options

A. 0.012
B. 0.013
C. 0.014
D. 0.015

Answer

To find the possible percentage error in computing the parallel resistance \( R \) of three resistances \( R_1, R_2, R_3 \) when each resistance has an error of \( +1.2\% \), we can follow these steps:

### Step 1: Write the formula for parallel resistance
The formula for the equivalent resistance \( R \) in parallel is given by:
\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

### Step 2: Define the percentage error
The percentage error in a quantity \( X \) is defined as:
\[ \text{Percentage Error} = \frac{\Delta X}{X} \times 100 \]
where \( \Delta X \) is the absolute error in \( X \).

### Step 3: Apply the error propagation formula
For the function \( \frac{1}{R} \), the error propagation formula states:
\[ \left(\frac{\Delta \left(\frac{1}{R}\right)}{\frac{1}{R}}\right)^2 = \left(\frac{\Delta R_1}{R_1}\right)^2 + \left(\frac{\Delta R_2}{R_2}\right)^2 + \left(\frac{\Delta R_3}{R_3}\right)^2 \]

### Step 4: Substitute the known percentage errors
Given that each resistance \( R_1, R_2, R_3 \) has a percentage error of \( +1.2\% \), we have:
\[ \frac{\Delta R_1}{R_1} = \frac{\Delta R_2}{R_2} = \frac{\Delta R_3}{R_3} = 0.012 \]

### Step 5: Calculate the total error in \( \frac{1}{R} \)
Substituting these values into the error propagation formula:
\[ \left(\frac{\Delta \left(\frac{1}{R}\right)}{\frac{1}{R}}\right)^2 = (0.012)^2 + (0.012)^2 + (0.012)^2 \]
\[ = 3 \times (0.012)^2 = 3 \times 0.000144 = 0.000432 \]

### Step 6: Take the square root to find the error in \( \frac{1}{R} \)
Now, taking the square root gives:
\[ \frac{\Delta \left(\frac{1}{R}\right)}{\frac{1}{R}} = \sqrt{0.000432} \approx 0.0208 \]

### Step 7: Convert to percentage error
To find the percentage error in \( R \), we note that:
\[ \frac{\Delta R}{R} = \left|\frac{\Delta \left(\frac{1}{R}\right)}{\frac{1}{R}}\right| \]
Thus, the percentage error in \( R \) is:
\[ \text{Percentage Error in } R = \frac{\Delta R}{R} \times 100 = 0.0208 \times 100 \approx 2.08\% \]

### Final Answer
The possible percentage error in computing the parallel resistance \( R \) is approximately \( 2.08\% \).

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