27. Resistors & DC Circuits

The voltage across the terminals of a 9.0 V battery is 8.5 V when the battery is connected to a 20 Ω load. What is the battery's internal resistance?

(II) How long does it take for the energy stored in a capacitor in a series RC circuit (Fig. 26–65) to reach 75% of its maximum value? Express answer in terms of the time constant $\tau =RC$.

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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?)

The capacitor in FIGURE P28.73 begins to charge after the switch closes at t = 0 s. Find an expression for the current I at time t. Graph I from t = 0 to t = 5τ.

The current supplied by a battery slowly decreases as the battery runs down. Suppose that the current as a function of time is $I=\left(0.75A\right)e^{-\frac{t}{6h}}$. What is the total number of electrons transported from the positive electrode to the negative electrode by the charge escalator from the time the battery is first used until it is completely dead?

[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,

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?

(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?

(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.

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?

Two 10-cm-diameter metal plates 1.0 cm apart are charged to ±12.5 nC. They are suddenly connected together by a 0.224-mm-diameter copper wire stretched taut from the center of one plate to the center of the other. Does the current increase with time, decrease with time, or remain steady? Explain.

[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 $\Delta P\approx \Delta R{\frac{dP}{dR}\mid }_{R=R_0}$]

Cardiac defibrillators are discussed in Section 24–4. Choose a value for the resistance so that the 1.4-μF capacitor can be charged to 3100 V in 2.0 seconds. Assume that this 3100 V is 95% of the full source voltage.

Neglect the internal resistance of a battery unless the Problem refers to it. Two resistors when connected in series to a 120-V line use one-fourth the power that is used when they are connected in parallel. If one resistor is 4.3 kΩ, what is the resistance of the other?

(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?