Question

What is the equivalent resistance of three resistors, each of resistance R, if they are connected to an ideal battery (a) in series with one another and (b) in parallel with one another? (c) Is the potential difference across the series arrangement greater than, less than, or equal to that across the parallel arrangement?

Short answer

1. Equivalent resistance for series arrangement is 3R
2. Equivalent resistance for parallel arrangement is R/3
3. Potential difference across series arrangement is the same as that of parallel arrangement.

Step-by-step solution

Step 1: Given

Each resistance is R.

Determining the concept

Here, use the formula for equivalent resistance for series as well as parallel arrangement.

Formulae are as follow:

$$\begin{gathered} R_{series}=R_1+R_2 \\ \frac{1}{R_{parallel}}=\frac{1}{R_1}+\frac{1}{R_2} \end{gathered}$$

Where, R is resistance.

(a) Determining the equivalent resistance for series arrangement

Equivalent resistance for series arrangement:

For series arrangement,

$$\begin{gathered} R_{series}=R+R+R \\ {R}_{series}=3R \end{gathered}$$

Hence, equivalent resistance for series arrangement is 3R.

(b) Determining the equivalent resistance for parallel arrangement

Equivalent resistance for series arrangement:

$$\begin{gathered} \frac{1}{R_{parallel}}=\frac{1}{R}+\frac{1}{R}+\frac{1}{R} \\ {R}_{parallel}=\frac{R}{3} \end{gathered}$$

Hence, equivalent resistance for parallel arrangement is R/3.

(c) Determining whether potential difference across series is greater, less, or equal to that of parallel arrangement

In series arrangement, potential difference across each resistance is less than battery voltage as it is equally distributed, and the total potential difference across all resistors connected in series is equal to the battery voltage.

In parallel arrangement, potential difference across each resistance is the same, and it is equal to battery voltage.

This means if the potential difference across complete arrangement is considered, then it is the same for both. But for an individual resistor, resistance in series arrangement has less potential difference than the resistor in parallel arrangement.

Hence, potential difference across series arrangement is the same as that of parallel arrangement.

Therefore, use the formula for equivalent resistance for series as well as parallel arrangement. From this, find the equivalent resistance.