Chapter 23, Problem 86A

To determine

To explain: If there is a need to buy a set of new resistors of a desired value or can they be obtained from a set of resistor given value.

Answer to Problem 86A

No, new resistors are not required since the required resistance value can be obtained by using specific combination of the available resistor.

Explanation of Solution

Given:

A set of 30 Ω resistors and a 10-Ω resistor and 15-Ω resistor are needed.

Two or more resistors can be connected in series or parallel configurations to obtain new values. The desired value of the equivalent resistance can be obtained by choosing the proper combination of resistances.

The fundamental idea is that a series combination of resistors would give an equivalent resistance value which is greater than the greatest valued resistor in the group. But the parallel combination of resistors gives you an equivalent resistance smaller than the smallest valued resistor

In the given case, a set of 30-Ω resistors are available. Since the desired resistors are less valued than the original resistance, a parallel combination is to be considered.

When n identical resistors are connected in parallel, the equivalent resistance is expressed as,

  $R_{eq}=\frac{R}{n}$

Here the resistance R =30-Ω. Let’s consider the first resistance required which is 10-Ω. This implies,

  $\begin{array}{l}10=\frac{30}{n}\\ n=3\end{array}$

This implies, a parallel combination of the three resistance each 30-Ω will give 10-Ω as the equivalent resistance.

Now, consider the case when n =2. This gives the equivalent resistance as,

  $\begin{array}{c}{R}_{eq}=\frac{30}{2}\\ =15\Omega \end{array}$

This implies a parallel combination of two resistors each 30-Ω will give 15-Ω as the equivalent resistance.

From the above analysis, it can be understood that the existing resistance can used to obtain the new resistance values.

Conclusion:

There is no need to buy 15-Ω and 10-Ω resistors separately. They can be obtained from the existing 30-Ω resistor.