Calculate equivalent resistance of two resistors `R_(1) and R_(2)` in parallel where, `R_(1) = (6+-0.2)`ohm and ` R_2 = (3+-0.1)ohm`

Explanation

To calculate the equivalent resistance of two resistors \( R_1 \) and \( R_2 \) in parallel, we will follow these steps:

### Step 1: Write the formula for equivalent resistance in parallel
The formula for the equivalent resistance \( R \) of two resistors \( R_1 \) and \( R_2 \) connected in parallel is given by:

\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 2: Substitute the values of \( R_1 \) and \( R_2 \)
Given:
- \( R_1 = 6 \, \Omega \) with an uncertainty of \( \pm 0.2 \, \Omega \)
- \( R_2 = 3 \, \Omega \) with an uncertainty of \( \pm 0.1 \, \Omega \)

Substituting these values into the formula:

\[ \frac{1}{R} = \frac{1}{6} + \frac{1}{3} \]

### Step 3: Calculate the right-hand side
Calculating the fractions:

\[ \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 4: Find the equivalent resistance \( R \)
Now, taking the reciprocal to find \( R \):

\[ R = \frac{1}{\frac{1}{2}} = 2 \, \Omega \]

### Step 5: Calculate the maximum permissible error
To find the maximum permissible error in the equivalent resistance, we differentiate the equation for \( R \):

\[ -\frac{dR}{R^2} = -\frac{dR_1}{R_1^2} - \frac{dR_2}{R_2^2} \]

Rearranging gives:

\[ dR = \left( \frac{dR_1}{R_1^2} + \frac{dR_2}{R_2^2} \right) R^2 \]

### Step 6: Substitute the uncertainties and values
Given:
- \( dR_1 = 0.2 \, \Omega \)
- \( dR_2 = 0.1 \, \Omega \)
- \( R_1 = 6 \, \Omega \)
- \( R_2 = 3 \, \Omega \)
- \( R = 2 \, \Omega \)

Substituting these values into the error formula:

\[ dR = \left( \frac{0.2}{6^2} + \frac{0.1}{3^2} \right) (2^2) \]

Calculating each term:

\[ \frac{0.2}{36} + \frac{0.1}{9} = \frac{0.2}{36} + \frac{0.4}{36} = \frac{0.6}{36} = \frac{1}{60} \]

Now, substituting back:

\[ dR = \left( \frac{1}{60} \right) (4) = \frac{4}{60} = \frac{1}{15} \approx 0.0667 \, \Omega \]

### Step 7: Final result with uncertainty
The equivalent resistance \( R \) with its uncertainty is:

\[ R = 2 \pm 0.07 \, \Omega \]

### Summary
The equivalent resistance of the two resistors in parallel is:

\[ R = 2 \pm 0.07 \, \Omega \]