A combination of two resisters `R_(1)` and `R_(4)` are placed in an electrical circuit. In an experimental arrangement, the resistance are measured as `R_(1)=(100 pm 3)Omega , R_(2)=(200 pm 4)Omega`
The equivalent resistance, when they are connected in parallel, `(1)/(R_(P))=(1)/(R_(1))+(1)/(R_(2))` can be expressed as

Options

A. `(66.7 pm 12)Omega`
B. `(66.7 pm 7)Omega`
C. `(66.7 pm 4.9)Omega`
D. `(6.7 pm 1.8)Omega`

Answer

To solve the problem of finding the equivalent resistance \( R_P \) when two resistors \( R_1 \) and \( R_2 \) are connected in parallel, we will follow these steps:

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

### Step 2: Substitute the values of \( R_1 \) and \( R_2 \)
Given:
- \( R_1 = 100 \pm 3 \, \Omega \)
- \( R_2 = 200 \pm 4 \, \Omega \)

Substituting these values into the formula:
\[ \frac{1}{R_P} = \frac{1}{100} + \frac{1}{200} \]

### Step 3: Calculate \( \frac{1}{R_P} \)
To calculate \( \frac{1}{R_P} \), we need a common denominator:
\[ \frac{1}{R_P} = \frac{2}{200} + \frac{1}{200} = \frac{2 + 1}{200} = \frac{3}{200} \]

### Step 4: Find \( R_P \)
Now, take the reciprocal to find \( R_P \):
\[ R_P = \frac{200}{3} \, \Omega \]

### Step 5: Calculate the error in \( R_P \)
To find the error in \( R_P \), we use the formula for the propagation of uncertainty:
\[ \Delta R_P = R_P \sqrt{\left(\frac{\Delta R_1}{R_1}\right)^2 + \left(\frac{\Delta R_2}{R_2}\right)^2} \]
Where:
- \( \Delta R_1 = 3 \, \Omega \)
- \( \Delta R_2 = 4 \, \Omega \)

Substituting the values:
\[ \Delta R_P = \frac{200}{3} \sqrt{\left(\frac{3}{100}\right)^2 + \left(\frac{4}{200}\right)^2} \]

### Step 6: Simplify the error calculation
Calculating the terms inside the square root:
\[ \left(\frac{3}{100}\right)^2 = \frac{9}{10000}, \quad \left(\frac{4}{200}\right)^2 = \left(\frac{1}{50}\right)^2 = \frac{1}{2500} = \frac{4}{10000} \]
Thus,
\[ \Delta R_P = \frac{200}{3} \sqrt{\frac{9 + 4}{10000}} = \frac{200}{3} \sqrt{\frac{13}{10000}} = \frac{200}{3} \cdot \frac{\sqrt{13}}{100} \]

Calculating \( \sqrt{13} \approx 3.605 \):
\[ \Delta R_P \approx \frac{200 \cdot 3.605}{300} \approx \frac{721}{300} \approx 2.403 \, \Omega \]

### Step 7: Final Result
Thus, the equivalent resistance with its uncertainty is:
\[ R_P = \frac{200}{3} \pm 2.403 \, \Omega \approx 66.67 \pm 2.40 \, \Omega \]

### Final Answer
The equivalent resistance \( R_P \) can be expressed as:
\[ R_P = 66.67 \pm 2.40 \, \Omega \]