Chapter 35, Problem 18A

The resistors in the circuits below are $10\Omega$ each. Each circuit is powered with a 12-V battery. Assume that the battery and connecting wires have negligible resistance.

a. Rank the circuits in terms of the amount of current passing point P in each circuit, from greatest current to least current.

b. Suppose a voltmeter is connected between point P and the negative terminal of the battery. Rank the circuits in terms of voltmeter readings, from highest voltage to lowest voltage.

To determine

Ranking of circuits in terms of current passing through the point P from greatest to least.

Answer to Problem 18A

A>B> C

Explanation of Solution

Given:

The voltage of the battery in each circuit, $V=12\text{V}$

The resistance of each resistor in the circuit, $R=10\Omega$

Formula Used:

The expression to calculate the equivalent resistance of the circuit A is,

$R_A=R$

The expression to calculate the current passing through point P in circuit A is,

$I_A=\frac{V}{R_A}$

The expression to calculate the equivalent resistance of the circuit B is,

$\begin{array}{c}{R}_B=R+R\\ =2R\end{array}$

The expression to calculate the current passing through point P in circuit B is,

$I_B=\frac{V}{R_B}$

The expression to calculate the equivalent resistance of the circuit C is,

$\begin{array}{c}{R}_C=R+R+R\\ =3R\end{array}$

The expression to calculate the current passing through point P in circuit B is,

$I_C=\frac{V}{R_C}$

Calculation:

Substitute values in the above expression.

$R_A=10\Omega$

Substitute values in the above expression.

$\begin{array}{c}{I}_A=\frac{12\text{V}}{10\Omega }\\ =1.2\text{A}\end{array}$

Substitute values in the above expression.

$\begin{array}{c}{R}_B=2\times 10\Omega \\ =20\Omega \end{array}$

Substitute values in the above expression.

$\begin{array}{c}{I}_B=\frac{12\text{V}}{20\Omega }\\ =0.6\text{A}\end{array}$

Substitute values in the above expression.

$\begin{array}{c}{R}_C=3\times 10\Omega \\ =30\Omega \end{array}$

Substitute values in the above expression.

$\begin{array}{c}{I}_C=\frac{12\text{V}}{30\Omega }\\ =0.4\text{A}\end{array}$

From the above, it is clear that the current at the point P in the circuit A is 1.2 A. The current in the circuit B is 0.6 A. The current in the circuit C is 0.4 A.

Conclusion:

Thus, the ranking of the circuit for the current at point P in each circuit, from greatest to lowest, is A, B, C.

To determine

Ranking of circuits in terms of voltage readings from the highest voltage to lowest voltage.

Answer to Problem 18A

C > B >A

Explanation of Solution

Introduction:

The voltage in the voltmeter depends on the resistance of the circuit. If resistance increases, then the voltagedrop increases.

The equivalent resistance in circuit C is the greatest.So, the reading of the voltmeter between the negative terminal of the battery and point P in the circuit is greatest for circuit C.

The equivalent resistance in circuit A is the least. So, the reading of the voltmeter between the negative terminal of the battery and point P in the circuit is least for circuit A.

The equivalent resistance of the circuit B lies between the equivalent resistances of circuit A and circuit C. So, the reading for the voltmeter also lies between the readings of circuit A and circuit C.

Conclusion:

Thus, the reading of the voltmeter in circuit C is greatest, the reading for circuit A is least and for circuit B, the reading lies between the readings of circuit A and circuit C.