Two resistances `4 pm 0.4 Omega` and ` 4 pm 0.8 Omega` are in parallel. Find the equivalent resistance?

Options

A. `2 pm 0.3 Omega`
B. `2 pm 0.4 Omega`
C. `4 pm 0.3 Omega`
D. `4 pm 0.4 Omega`

Answer

To find the equivalent resistance of two resistors in parallel, we can use the formula:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

Given:
- \( R_1 = 4 \pm 0.4 \, \Omega \)
- \( R_2 = 4 \pm 0.8 \, \Omega \)

### Step 1: Calculate the nominal equivalent resistance

1. Calculate the nominal values of \( R_1 \) and \( R_2 \):
- \( R_1 = 4 \, \Omega \)
- \( R_2 = 4 \, \Omega \)

2. Substitute these values into the formula for equivalent resistance:
\[ \frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{4} \]
\[ \frac{1}{R_{eq}} = \frac{2}{4} = \frac{1}{2} \]

3. Therefore, the nominal equivalent resistance \( R_{eq} \) is:
\[ R_{eq} = 2 \, \Omega \]

### Step 2: Calculate the uncertainty in equivalent resistance

To find the uncertainty in the equivalent resistance, we differentiate the formula for \( R_{eq} \):

1. Differentiate the equation:
\[ dR_{eq} = R_{eq}^2 \left( \frac{dR_1}{R_1^2} + \frac{dR_2}{R_2^2} \right) \]

2. Substitute the values:
- \( R_{eq} = 2 \, \Omega \)
- \( dR_1 = 0.4 \, \Omega \)
- \( dR_2 = 0.8 \, \Omega \)
- \( R_1 = 4 \, \Omega \)
- \( R_2 = 4 \, \Omega \)

3. Calculate:
\[ dR_{eq} = 2^2 \left( \frac{0.4}{4^2} + \frac{0.8}{4^2} \right) \]
\[ = 4 \left( \frac{0.4}{16} + \frac{0.8}{16} \right) \]
\[ = 4 \left( \frac{0.4 + 0.8}{16} \right) \]
\[ = 4 \left( \frac{1.2}{16} \right) = 4 \times 0.075 = 0.3 \, \Omega \]

### Final Result

The equivalent resistance with uncertainty is:
\[ R_{eq} = 2 \pm 0.3 \, \Omega \]