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?

In Fig. 27-8a, calculate the potential difference between a and c by considering a path that contains R, r₁, and${\mathit{\epsilon }}_{\mathbf{1}}$ .

After the switch in Fig. 27-15 is closed on point a, there is current i through resistance R. Figure 27-23 gives that current for four sets of values of R and capacitance C: (1) ${\mathbf{R}}_{\mathbf{0}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{C}}_{\mathbf{0}}$, (2) $\mathbf{2}{\mathbf{R}}_{\mathbf{0}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{C}}_{\mathbf{0}}$, (3) ${\mathbf{R}}_{\mathbf{0}}\mathbf{}\mathbf{and}\mathbf{}\mathbf{2}{\mathbf{C}}_{\mathbf{0}}$, (4) $\mathbf{2}{\mathbf{R}}_{\mathbf{0}}\mathbf{}\mathbf{and}\mathbf{}\mathbf{2}{\mathbf{C}}_{\mathbf{0}}$. Which set goes with which curve?

Figure 27-63 shows an ideal battery of emf e= 12V, a resistor of resistance$\mathbf{R}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{\Omega }$, and an uncharged capacitor of capacitance $\mathbf{C}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{\mu F}$ . After switch S is closed, what is the current through the resistor when the charge on the capacitor $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{\mu C}$?

(a), both batteries have emf 1.20 V and the external resistance R is a variable resistor. Figure

(b) gives the electric potentials V between the terminals of each battery as functions of R: Curve 1 corresponds to battery 1, and curve 2 corresponds to battery 2.The horizontal scale is set by ${\mathit{R}}_{\mathbf{S}}\mathbf{=}$0.20 Ω. What is the internal resistance of (a) Battery 1 and

(b) Battery 2?

In Figure,${\mathit{R}}_{\mathbf{1}}\mathbf{=}\mathbf{100}\mathbf{\text{\hspace{0.17em}Ω}}$ , ${\mathit{R}}_{\mathbf{2}}\mathbf{=}\mathbf{50}\mathbf{\text{\hspace{0.17em}Ω}}$, and the ideal batteries have emfs $\mathbf{}{\mathit{\epsilon }}_{\mathbf{1}}\mathbf{=}\mathbf{6}\mathbf{.0}\mathbf{\text{\hspace{0.17em}V}}$,$\mathbf{}{\mathit{\epsilon }}_{\mathbf{2}}\mathbf{=}\mathbf{5}\mathbf{.0}\mathbf{\text{\hspace{0.17em}V}}$ , and .${\mathit{\epsilon }}_{\mathbf{3}}\mathbf{=}\mathbf{4}\mathbf{.0}\mathbf{\text{\hspace{0.17em}V}}$ Find

(a) The current in resistor 1,

(b) The current in resistor 2, and

(c) The potential difference between points a and b.