Chapter 27: Circuits

The starting motor of a car is turning too slowly, and the mechanic has to decide whether to replace the motor, the cable, or the battery. The car’s manual says that the12Vbattery should have no more than$\mathbf{0}\mathbf{.}\mathbf{020}\mathbf{\Omega }$ internal resistance; the motor should have no more than $\mathbf{0}\mathbf{.}\mathbf{200}\mathit{\Omega }$resistance, and the cable no more than $\mathbf{0}\mathbf{.}\mathbf{040}\mathit{\Omega }$resistance. The mechanic turns on the motor and measures 11.4Vacross the battery, a 3.0Vcross the cable, and a current of 50A. Which part is defective?

Question: Two resistors $R_{1}and{R}_2$ may be connected either in series or in parallel across an ideal battery with emf $\epsilon$. We desire the rate of energy dissipation of the parallel combination to be five times that of the series combination. Iflocalid="1662972116950" $R_1=100\Omega$, what are the (a) smaller and (b) larger of the two values of$R_2$that result in that dissipation rate?

The circuit of Fig. 27-75 shows a capacitor, two ideal batteries, two resistors, and a switch S. Initially S has been open for a long time. If it is then closed for a long time, what is the change in the charge on the capacitor? Assume,$\mathbf{C}\mathbf{=}\mathbf{10}\mathbf{\mu F}$,${\mathbf{\epsilon }}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{V}$ ,${\mathbf{\epsilon }}_{\mathbf{1}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{V}$ , ${\mathbf{R}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{\Omega }$and${\mathbf{R}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{40}\mathbf{\Omega }$.

In Fig. 27-41,${\mathbf{R}}_{\mathbf{1}}\mathbf{=}\mathbf{10}\mathbf{\Omega }$,${\mathbf{R}}_{\mathbf{2}}\mathbf{=}\mathbf{20}\mathbf{\Omega }$, and the ideal batteries have emfs ${\mathbf{\epsilon }}_{\mathbf{1}}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{V}$and ${\mathbf{\epsilon }}_{\mathbf{2}}\mathbf{=}\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{V}$. What value of results in no current through battery 1?

In Fig. 27-76,$\mathit{R}\mathbf{=}\mathbf{10}\mathit{\Omega }$. What is the equivalent resistance between points A and B? (Hint: This circuit section might look simpler if you first assume that points A and B are connected to a battery.)

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?

Question: (a) In Fig. 27-4a, show that the rate at which energy is dissipated in Ras thermal energy is a maximum when R =r. (b) Show that this maximum power is $P={\epsilon }^2/4r$.

Question: In Fig. 27-77, the ideal batteries have emfs ${\epsilon }_1=12.0Vand{\epsilon }_2=4.0V$, and the resistances are each$4.00\Omega$. What are the (a) size and (b) direction (up or down) of$i_1$and the (c) size and (d) direction of$i_2$? (e) Does battery 1 supply or absorb energy, and (f) what is its energy transfer rate? (g) Does battery 2 supply or absorb energy, and (h) what is its energy transfer rate?

Figure 27-78 shows a portion of a circuit through which there is a current I=6.00A. The resistances are R₁.=R₂=2.00R₃=2.OOR₄=4.00$\Omega$What is the currenti₁through resistor 1?

Thermal energy is to be generated in a$\mathbf{0}\mathbf{.10}\mathbf{}\mathbf{\text{\hspace{0.17em}Ω}}$resistor at the rate of$\mathbf{10}\mathbf{}\mathbf{\text{W}}$by connecting the resistor to a battery whose emf is$\mathbf{1}\mathbf{.5}\mathbf{}\mathbf{\text{V}}$. (a) What potential difference must exist across the resistor? (b) What must be the internal resistance of the battery?