27. Resistors & DC Circuits

(III) If the 25-Ω resistor in Fig. 26–59 is shorted out (resistance = 0 ), what then would be the current through the 15-Ω resistor?

Cardiac defibrillators are discussed in Section 24–4. The effective resistance of the human body is given in Section 26–6. If the defibrillator discharges with a time constant of 12 ms, what is the effective capacitance of the human body?

A typical voltmeter has an internal resistance of 10 MΩ and can only measure potential differences of up to several hundred volts. Figure 26–93 shows the design of a probe to measure a very high voltage V using a voltmeter. If you want the voltmeter to read 50 V when V = 50 kV , what value R should be used in this probe?

The current through the 4.0-kΩ resistor in Fig. 26–74 is 2.85 mA. What is the terminal voltage V_ba of the “unknown” battery? (There are two answers. Why?)

Suppose the switch S in Fig. 26–70 is closed. What is the time constant (or time constants) for charging the capacitors after the 24 V is applied?

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]

(II) Determine the equivalent resistance of the circuit shown in Fig. 26–44,

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]

(II) Determine the voltage across each resistor

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]

(II) Determine the current through each resistor.

The level of liquid helium (temperature ≈ 4K) in its storage tank can be monitored using a vertically aligned niobium–titanium (NbTi) wire, whose length ℓ spans the height of the tank. In this level-sensing setup, an electronic circuit maintains a constant electrical current I at all times in the NbTi wire and a voltmeter monitors the voltage V across this wire. The NbTi wire is superconducting ( R = 0) if below its transition temperature of 10 K, so the portion of the wire immersed in the liquid helium is in the superconducting state, while the portion above the liquid (in helium vapor with temperature above 10 K) is in the normal state. Define ƒ = x/ℓ to be the fraction of the tank filled with liquid helium (Fig. 25–40) and V₀ to be the value of the voltage V when the tank is empty (ƒ = 0) . Determine the relation between f and V (in terms of V₀).

[In these Problems neglect the internal resistance of a battery unless the Problem refers to it.]

(III) You are designing a wire resistance heater to heat an enclosed container of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value R = R₀, but may fluctuate slightly causing its resistance to vary a small amount ∆R ( << R₀ ). To maintain the heater at constant power, you design the circuit shown in Fig. 26–50, which includes two resistors, each of resistance R′. Determine the value for R′ so that the heater power P will remain constant even if its resistance R fluctuates by a small amount. [Hint: If ∆R << R₀ , then $\Delta P\approx \Delta R{\frac{dP}{dR}\mid }_{R=R_0}$]

(II) Determine the magnitudes and directions of the currents in each resistor shown in Fig. 26–57. The batteries have emfs of ε₁ = 9.0V and ε₂ = 12.0V and the resistors have values of R₁ = 25 Ω, R₂ = 48 Ω, and R₃ = 35 Ω.

(a) Ignore internal resistance of the batteries.

(b) Assume each battery has internal resistance r = 1.0 Ω.

(II) Suppose two batteries, with unequal emfs of 2.00 V and 3.00 V, are connected as shown in Fig. 26–63. If each internal resistance is r = 0.350Ω and R = 4.00Ω, what is the voltage across the resistor R?

(III) (a) Determine the currents I₁, I₂, and I₃ in Fig. 26–58. Assume the internal resistance of each battery is r = 1.0 Ω.

(b) What is the terminal voltage of the 6.0-V battery?

The circuit shown in Fig. 26–89 is a primitive 4-bit digital-to-analog converter (DAC). In this circuit, to represent each digit (2ⁿ) of a binary number, a “1” has the nᵗʰ switch closed whereas zero (“0”) has the switch open. For example, 0010 is represented by closing switch n = 1, while all other switches are open. Show that the voltage V across the 1.0 - Ω resistor for the binary numbers 0001, 0010, 0100, and 1010 (which represent 1, 2, 4, 10) follows the pattern that you expect for a 4-bit DAC.

(II) (a) What is the potential difference between points a and d in Fig. 26–55 (similar to Fig. 26–12, Example 26–8), and (b) what is the terminal voltage of each battery?