The effective resistance of a circuit containing resistances in parallel is

Options

A. equal to the sum of the individual resistances
B. smaller than any of the individual resistances
C. greater than any of the individual resistances
D. sometimes greater and sometimes smaller than the individual resistances

Answer

To find the effective resistance of a circuit containing resistances in parallel, we can follow these steps:

### Step 1: Understand the Concept of Parallel Resistance
In a parallel circuit, the voltage across each resistor is the same, and the total current is the sum of the currents through each resistor. The effective resistance (R_eq) in a parallel circuit is always less than the smallest individual resistance.

### Step 2: Use the Formula for Effective Resistance
The formula for calculating the effective resistance (R_eq) of resistors in parallel is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \]

Where \( R_1, R_2, R_3, \ldots \) are the individual resistances.

### Step 3: Analyze the Options
Given the options:
1. Equal to the sum of the individual resistances
2. Smaller than any of the individual resistances
3. Greater than any of the individual resistances
4. Sometimes greater and sometimes smaller than the individual resistance

### Step 4: Evaluate Each Option
- **Option 1**: This is incorrect because in a parallel combination, the effective resistance is not the sum of the individual resistances; that applies to series combinations.

- **Option 2**: This is correct. The effective resistance in a parallel circuit is always less than the smallest individual resistance.

- **Option 3**: This is incorrect because the effective resistance cannot be greater than any of the individual resistances in a parallel circuit.

- **Option 4**: This is also incorrect because the effective resistance in a parallel circuit is consistently less than the individual resistances, not variable.

### Step 5: Conclusion
Thus, the effective resistance of a circuit containing resistances in parallel is always smaller than any of the individual resistances. Therefore, the correct answer is **Option 2**.

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