The total resistance in a parallel circuit is less than the smallest of the individual resistances.a.Falseb.True

Step 1: Break Down the Problem

To evaluate whether the statement "The total resistance in a parallel circuit is less than the smallest of the individual resistances" is true or false, we need to consider the rules pertaining to resistances in a parallel circuit.

Step 2: Relevant Concepts

In a parallel circuit, the formula for total resistance $R_t$ is given by:

$\frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

where $R_1, R_2, R_3, \ldots$ are the individual resistances. The total resistance in a parallel setup can be compared to the smallest resistance $R_{min}$.

Step 3: Analysis and Detail

1. Understanding the Formula: Since all resistances are positive, the terms on the right-hand side of the equation will be positive fractions. Therefore, as more resistances are added in parallel, $\frac{1}{R_t}$ increases, which results in $R_t$ decreasing.

2. Comparison of Resistances:

   • Assume we have resistances $R_1, R_2, R_3$ with $R_{min} = \min(R_1, R_2, R_3)$.
   • For any individual resistance $R_i$: $\frac{1}{R_t} > \frac{1}{R_{min}} \implies R_t < R_{min}$
   • Hence, $R_t$ must be less than the smallest individual resistor.

Step 4: Verify and Summarize

The derived conclusion shows that the total resistance $R_t$ in a parallel circuit indeed is less than the smallest of the individual resistances. This is confirmed by the mathematical principles governing resistances in parallel circuits.

Final Answer

b. True. The total resistance in a parallel circuit is less than the smallest of the individual resistances.