Question

A total resistance of 3.00 Ω is to be produced by connecting an unknown resistance to a 12.0 Ω resistance.

1. What must be the value of the unknown resistance, and
2. (b) Should it be connected in series or in parallel?

Short answer

1. The value of the unknown resistance is$R^{\text{'}}=4.00\text{Ω}$
2. The unknown resistance should be connected in parallel.

Step-by-step solution

Given

$$\begin{gathered} R_{eq}=3.00\text{Ω} \\ R=12.0\text{Ω} \end{gathered}$$

Determining the concept

It can be predicted whether the unknown resistance is connected parallel or series from the given values. Then using the formula for corresponding equivalent resistance, findthe value of the unknown resistance.

Formulae are as follow:

$\frac{1}{R_{eq}}=\frac{1}{R}+\frac{1}{R^{\text{'}}}$

Where, R is resistance.

(a) determining the value of the unknown resistance

Let the unknown resistance connected be R’.

From the given values, we can write that$R_{eq}<R.$

This implies that the unknown resistance is connected in parallel.

Hence,

$$\begin{gathered} \frac{1}{R_{eq}}=\frac{1}{R}+\frac{1}{R^{\text{'}}} \\ \therefore R_{eq}=\frac{R{R}^{\text{'}}}{R+R^{\text{'}}} \\ 3=\frac{12R^{\text{'}}}{12+R^{\text{'}}} \\ 36+3R\text{'}=12R\text{'} \\ {R}^{\text{'}}=4.00\text{Ω} \end{gathered}$$

Hence, the value of the unknown resistance is$R^{\text{'}}=4.00\text{Ω}$

(a) Determining the unknown resistance should be connected in series or in parallel

From part a),

it can be concluded that the unknown resistance should be connected in parallel.

Hence, the unknown resistance should be connected in parallel.

Therefore, by using the formula for corresponding equivalent resistance, unknown resistance can be determined.