The resistance R of a wire is found by determining its length l and radius r. The percentage errors in measurement of l and r are respectively 1% and 2%. The percentage error in measurement of R is

Options

A. `4%`
B. `5%`
C. `3%`
D. `1,5%`

Answer

To find the percentage error in the measurement of resistance \( R \) of a wire based on the given percentage errors in its length \( l \) and radius \( r \), we can follow these steps:

### Step 1: Understand the formula for resistance
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho l}{A} \]
where \( A \) is the cross-sectional area of the wire. For a circular wire, the area \( A \) can be expressed in terms of the radius \( r \):
\[ A = \pi r^2 \]
Thus, we can rewrite the resistance as:
\[ R = \frac{\rho l}{\pi r^2} \]

### Step 2: Identify the variables and their errors
From the problem, we know:
- The percentage error in the measurement of length \( l \) is \( 1\% \).
- The percentage error in the measurement of radius \( r \) is \( 2\% \).

### Step 3: Apply the formula for percentage error
The formula for the percentage error in a product or quotient of variables is given by:
\[ \text{Percentage Error in } R = \text{Percentage Error in } l + 2 \times \text{Percentage Error in } r \]
This is because the radius \( r \) appears squared in the formula for resistance.

### Step 4: Substitute the known values
Now, substituting the known percentage errors:
\[ \text{Percentage Error in } R = 1\% + 2 \times 2\% \]
Calculating this gives:
\[ \text{Percentage Error in } R = 1\% + 4\% = 5\% \]

### Conclusion
The percentage error in the measurement of resistance \( R \) is \( 5\% \).

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