For two resistors `R_1` and `R_2`, connected in parallel, find the relative error in their equivalent resistance. if `R_1=(50 pm2)Omega ` and `R_2=(100 pm 3) Omega`.

Answer

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

### Step 1: Understand the formula for equivalent resistance in parallel
The equivalent resistance \( R_{\text{eq}} \) for two resistors \( R_1 \) and \( R_2 \) connected in parallel is given by the formula:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} \]
This can be rearranged to:
\[ R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2} \]

### Step 2: Substitute the values of \( R_1 \) and \( R_2 \)
Given:
- \( R_1 = 50 \pm 2 \, \Omega \)
- \( R_2 = 100 \pm 3 \, \Omega \)

Substituting the nominal values into the formula:
\[ R_{\text{eq}} = \frac{50 \times 100}{50 + 100} = \frac{5000}{150} = \frac{100}{3} \, \Omega \approx 33.33 \, \Omega \]

### Step 3: Differentiate the equation for error propagation
To find the relative error in \( R_{\text{eq}} \), we use the formula for error propagation in functions of multiple variables. The differential form gives us:
\[ \delta R_{\text{eq}} = R_{\text{eq}}^2 \left( \frac{\delta R_1}{R_1^2} + \frac{\delta R_2}{R_2^2} \right) \]
Where \( \delta R_1 \) and \( \delta R_2 \) are the uncertainties in \( R_1 \) and \( R_2 \).

### Step 4: Substitute the uncertainties and values into the equation
We know:
- \( \delta R_1 = 2 \, \Omega \)
- \( \delta R_2 = 3 \, \Omega \)

Now substituting into the error propagation equation:
\[ \delta R_{\text{eq}} = \left( \frac{100}{3} \right)^2 \left( \frac{2}{50^2} + \frac{3}{100^2} \right) \]

Calculating each term:
\[ \delta R_{\text{eq}} = \left( \frac{100}{3} \right)^2 \left( \frac{2}{2500} + \frac{3}{10000} \right) \]
\[ = \left( \frac{100}{3} \right)^2 \left( \frac{8}{20000} \right) \]
\[ = \left( \frac{10000}{9} \right) \left( \frac{8}{20000} \right) = \frac{80000}{180000} = \frac{8}{18} \approx 0.4444 \, \Omega \]

### Step 5: Calculate the relative error
The relative error in the equivalent resistance is given by:
\[ \text{Relative Error} = \frac{\delta R_{\text{eq}}}{R_{\text{eq}}} \]
Substituting the values:
\[ \text{Relative Error} = \frac{0.4444}{33.33} \approx 0.0133 \]

### Step 6: Final result
To express the relative error as a percentage:
\[ \text{Relative Error} \approx 0.0133 \times 100 \approx 1.33\% \]

Thus, the relative error in the equivalent resistance is approximately **1.33%**.

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