Question

State whether the following statement is true or false, and briefly explain why: “In the two circuits shown in Figure 19.64, the battery output power is greater in circuit 2 because there is an additional resistor dissipating power.”

Short answer

The statement is false since output power is inversely proportional to the total resistance of the circuit.

Step-by-step solution

Given data

A $10\text{V}$battery is connected first to a resistance $R_1$ (circuit 1) and then to a series connection of $R_1$ and $R_2$ (circuit 2).

The following statement is provided:

In the two circuits shown in Figure 19.64, the battery output power is greater in circuit 2 because there is an additional resistor $R_2$ dissipating power.

Dissipated power

The power dissipated in a circuit of net EMF and total resistance is

$\mathit{P}\mathbf{=}\frac{{\mathbf{E}}^{\mathbf{2}}}{\mathbf{R}}$ .....(i)

Determination of whether the provided statement is true is false

From equation (1), the total output power in circuit 1 is:

$P_1=\frac{100{\text{V}}^{\text{2}}}{R_1}$

The equivalent resistance of circuit 2 is $R_1+R_2$. Thus, from equation (1), the total output power in circuit 2 is:

$P_2=\frac{100{\text{V}}^{\text{2}}}{R_1+R_2}$

Evidently, $P_2>P_1$.

Thus, the statement is false.