Question

What is the equivalent resistance of two parallel resistors?

Answer 1

To calculate the equivalent resistance of two parallel resistors, you can use the formula:

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

Where \( R_1 \) and \( R_2 \) are the resistances of the two resistors. To find \( R_{eq} \), you take the reciprocal of the sum of the reciprocals of the individual resistances. After computing the sum, take the reciprocal of that result to find the equivalent resistance. If you have specific values for \( R_1 \) and \( R_2 \), I can help you calculate it!

Answer 2

The equivalent resistance of two resistors connected in parallel can be calculated using a specific formula. When resistors are connected in parallel, the total or equivalent resistance decreases. This is because the current has multiple paths to flow through, reducing the overall resistance.

### Formula for Two Parallel Resistors

If two resistors, \( R_1 \) and \( R_2 \), are connected in parallel, the formula for the equivalent resistance \( R_{\text{eq}} \) is given by:

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

Alternatively, you can rearrange this equation to find \( R_{\text{eq}} \):

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

### Explanation

1. **Reciprocal Relationship**: The formula involves the reciprocals of the resistances because, in a parallel circuit, the total conductance (inverse of resistance) is the sum of the individual conductances. The combined conductance of the circuit increases, resulting in a lower overall resistance.

2. **Derivation of the Formula**:
   - The conductance of a resistor \( R \) is \( \frac{1}{R} \).
   - The total conductance of two resistors in parallel is \( \frac{1}{R_1} + \frac{1}{R_2} \).
   - The equivalent resistance is the reciprocal of the total conductance, which leads to the formula above.

3. **Special Case - Identical Resistors**: If \( R_1 = R_2 = R \), the formula simplifies to:

\[
R_{\text{eq}} = \frac{R \times R}{R + R} = \frac{R^2}{2R} = \frac{R}{2}
\]

This means the equivalent resistance of two identical resistors in parallel is half of the resistance of one resistor.

### Example Calculation

Let's calculate the equivalent resistance of two resistors with values \( R_1 = 4 \, \Omega \) and \( R_2 = 6 \, \Omega \):

1. Use the formula:

\[
R_{\text{eq}} = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{4 \times 6}{4 + 6}
\]

2. Calculate:

\[
R_{\text{eq}} = \frac{24}{10} = 2.4 \, \Omega
\]

### Conclusion

The equivalent resistance of two parallel resistors is always less than the smallest resistor value among them. This property makes parallel circuits particularly useful in reducing the overall resistance in electrical networks.