Two resistance `R_(1)` and `R_(2)` are connected in (i) series and (ii) parallel. What is the equivalent resistance with limit of possible percentage error in each case of `R_(1)=5.0+-0.2Omega` and `R_(2)=10.0+-0.1Omega`

Answer

To solve the problem, we need to calculate the equivalent resistance of two resistors \( R_1 \) and \( R_2 \) in both series and parallel configurations, along with the limits of possible percentage error for each case.

### Given:
- \( R_1 = 5.0 \pm 0.2 \, \Omega \)
- \( R_2 = 10.0 \pm 0.1 \, \Omega \)

### Step 1: Calculate Equivalent Resistance in Series

1. **Formula for Series Resistance:**
\[ R_{\text{eq, series}} = R_1 + R_2 \]

2. **Substituting the values:**
\[ R_{\text{eq, series}} = 5.0 + 10.0 = 15.0 \, \Omega \]

3. **Calculating the error in series:**
\[ \Delta R_{\text{eq, series}} = \Delta R_1 + \Delta R_2 = 0.2 + 0.1 = 0.3 \, \Omega \]

4. **Final Result for Series:**
\[ R_{\text{eq, series}} = 15.0 \pm 0.3 \, \Omega \]

### Step 2: Calculate Equivalent Resistance in Parallel

1. **Formula for Parallel Resistance:**
\[ R_{\text{eq, parallel}} = \frac{R_1 R_2}{R_1 + R_2} \]

2. **Substituting the values:**
\[ R_{\text{eq, parallel}} = \frac{5.0 \times 10.0}{5.0 + 10.0} = \frac{50.0}{15.0} = \frac{10}{3} \, \Omega \approx 3.33 \, \Omega \]

3. **Calculating the error in parallel:**
- We use the formula for error in parallel resistances:
\[ \frac{\Delta R_{\text{eq, parallel}}}{R_{\text{eq, parallel}}} = \frac{\Delta R_1}{R_1} + \frac{\Delta R_2}{R_2} + \frac{\Delta R_s}{R_s} \]
Where \( R_s \) is the equivalent resistance in series, which we calculated earlier as 15.0 Ω and \( \Delta R_s = 0.3 \, \Omega \).

4. **Substituting the errors:**
\[ \frac{\Delta R_{\text{eq, parallel}}}{\frac{10}{3}} = \frac{0.2}{5.0} + \frac{0.1}{10.0} + \frac{0.3}{15.0} \]
\[ = 0.04 + 0.01 + 0.02 = 0.07 \]

5. **Calculating \( \Delta R_{\text{eq, parallel}} \):**
\[ \Delta R_{\text{eq, parallel}} = 0.07 \times \frac{10}{3} = \frac{0.7}{3} \, \Omega \approx 0.233 \, \Omega \]

6. **Final Result for Parallel:**
\[ R_{\text{eq, parallel}} = \frac{10}{3} \pm \frac{0.7}{3} \, \Omega \approx 3.33 \pm 0.23 \, \Omega \]

### Summary of Results:
- **Series:**
\[ R_{\text{eq, series}} = 15.0 \pm 0.3 \, \Omega \]
- **Parallel:**
\[ R_{\text{eq, parallel}} = \frac{10}{3} \pm \frac{0.7}{3} \, \Omega \approx 3.33 \pm 0.23 \, \Omega \]