Chapter 23.1, Problem 4PP

a.

To determine

Change inequivalent resistance of the circuit with three resistors in series, when one of the resistance value is changed.

Answer to Problem 4PP

The equivalent resistance of the circuit increases.

Explanation of Solution

Given:

Three resistors are connected in series across a 9-V battery and the resistance of one of the resistors is increased.

Formula used:

For a set of three resistors $R_1$ , $R_2$ , $R_3$ in series, the equivalent resistance can be calculated as,

  $R_{eq}=R_1+R_2+R_3$ …… (1)

  

Figure 1

Calculation:

Now, when the resistance of one of the resistors, say R₁ is increased by a value $\Delta R_1$ , it increases the effective resistance by the same value such that,

  $R_{eqnew}=R_1+\Delta R_1+R_2+R_3$

Conclusion:

The equivalent resistance of the series circuit increases when one of the resistance is increased.

b.

To determine

The change in current in the circuit with a given change in resistance in a series circuit

Answer to Problem 4PP

The current through the circuit decreases with increase in resistance of one of the resistors.

Explanation of Solution

Given:

Three resistors are connected in series across a 9-V battery and the resistance of one of the resistors is increased.

Formula used:

For the resistance of the circuit and supply voltage, the current can be calculated using the Ohm’s law as,

  $\begin{array}{l}V=IR\\ I=\frac{V}{R}\end{array}$ …… (2)

Calculation:

From the above relation, it is evident that current is inversely proportional to resistance.

In the initial case, the current in the circuit is given by,

  $I=\frac{V}{R_{eq}}$

Since the equivalent resistance has increased due to increase in one of the resistance values, the current decreases accordingly given by,

  $I_{new}=\frac{V}{R_{eqnew}}$

Conclusion:

The current through the series circuit decrease when one of the resistances is increased.

c.

To determine

To identify: Whether there will be change in the supply voltage for the chnage in resistance.

Answer to Problem 4PP

The supply voltage remains unchanged at the value of 9-V

Explanation of Solution

Given:

Three resistors are connected in series across a 9-V battery and the resistance of one of the resistors is increased.

Formula used:

The supply voltage of the circuit is expressed as,

  $V=I{R}_{eq}$

Calculation:

When one of the resistance values is increased, the current decreases accordingly. This increases the voltage drop across that particular resistor. But, the current in other resistors increases proportionately so that the total voltage remains constant, given by,

  $V=I_{new}{R}_{eqnew}$

This implies that, though the individual voltage and current distribution changes, the total voltage remains same.

Hence, the supply voltage remains unchanged.

Conclusion:

The supply voltage remains unchanged at 9-V even when resistance of one of the resistance value is increased