27. Resistors & DC Circuits

The circuit shown in Fig. E$25.33$ contains two batteries, each with an emf and an internal resistance, and two resistors. Find the terminal voltage $V_{ab}$ of the $16.0$-V battery.

The circuit shown in Fig. E$25.33$ contains two batteries, each with an emf and an internal resistance, and two resistors. Find the current in the circuit (magnitude and direction).

In the circuit shown in Fig. E26.27 find the current in the 3.00 Ω resistor.

Find the emfs ${\epsilon }_1$ and ${\epsilon }_2$ in the circuit of Fig. E26.26, and find the potential difference of point $b$ relative to point $a$.

(III) Determine the time constant for charging the capacitor in the circuit of Fig. 26–69. [Hint: Use Kirchhoff’s rules.] What is the maximum charge on the capacitor?

(I) Calculate the current in the circuit of Fig. 26–51, and show that the sum of all the voltage changes around the circuit is zero.

A potentiometer is a device to precisely measure potential differences or emf, using a “null” technique. In the simple potentiometer circuit shown in Fig. 26–83, R′ represents the total resistance of the resistor from A to B (which could be a long uniform “slide” wire), whereas R represents the resistance of only the part from A to the movable contact at C. When the unknown emf to be measured, $E_x$, is placed into the circuit as shown, the movable contact C is moved until the galvanometer G gives a null reading (i.e., zero) when the switch S is closed. The resistance between A and C for this situation we call R_x. Next, a standard emf, $E_s$, which is known precisely, is inserted into the circuit in place of $E_x$ and again the contact C is moved until zero current flows through the galvanometer when the switch S is closed. The resistance between A and C now is called R_s. Show that the unknown emf is given by $E_x=\left(\frac{R_x}{R_s}\right)E_s$ where R_x, R_s and $E_s$ are all precisely known. The working battery (at top of circuit diagram), as well as the standard cell $E_s$, are assumed to be fresh and to give a constant voltage.

(III) When the resistor R in Fig. 26–72 is 45 Ω, the high-resistance voltmeter reads 9.7 V. When R is replaced by a 14.0-Ω resistor, the voltmeter reading drops to 8.1 V. What are the emf and internal resistance of the battery?