27. Resistors & DC Circuits

What is current and voltage across each resistor below?

What is the voltage of the battery below?

(II) A galvanometer has a sensitivity of 45kΩ/V and internal resistance 20.0 Ω. How could you make this into

(a) an ammeter that reads 1.0 A full scale, or

(II) A galvanometer has an internal resistance of 32 Ω and deflects full scale for a 48-μA current. Describe how to use this galvanometer to make

(b) a voltmeter to give a full scale deflection of 250 V.

(II) A milliammeter reads 25 mA full scale. It consists of a 0.20-Ω resistor in parallel with a 33-Ω galvanometer. How can you change this ammeter to a voltmeter giving a full scale reading of 25 V without taking the ammeter apart? What will be the sensitivity (Ω/V) of your voltmeter?

(II) A 12.0-V battery (assume the internal resistance = 0) is connected to two resistors in series. A voltmeter whose internal resistance is 18.0 kΩ measures 5.5 V and 4.0 V, respectively, when connected across each of the resistors in turn. What is the resistance of each resistor?

Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire that when undeformed has length ℓ₀ , cross-sectional area A₀, and resistance R₀. This sensor is rigidly affixed to the object’s surface, aligning its length in the direction in which length changes are to be measured. As the object deforms, the length of the wire sensor changes by Δℓ, and the resulting change ΔR in the sensor’s resistance is measured. Assuming that as the solid wire is deformed to a length ℓ, its density and volume remain constant (only approximately valid), show that the strain ( = Δℓ / ℓ₀ ) of the wire sensor, and thus of the object to which it is attached, is approximately ΔR / 2R₀ .

(II) A parallel-plate capacitor is filled with a dielectric of dielectric constant K and high resistivity ρ (it conducts very slightly). This capacitor can be modeled as a pure capacitance C in parallel with a resistance R. Assume a battery places a charge +Q and -Q on the capacitor’s opposing plates and is then disconnected. Show that the capacitor discharges with a time constant τ = K ∊₀ρ (known as the dielectric relaxation time). Evaluate τ if the dielectric is glass with ρ = 1.0 x 10¹² Ω•m and K = 5.0.

Cardiac defibrillators are discussed in Section 24–4. (a) 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.

A 10.0-m length of wire consists of 5.0 m of copper followed by 5.0 m of aluminum, both of diameter 1.4 mm. A voltage difference of 75 mV is placed across the composite wire. (a) What is the total resistance (sum) of the two wires? (b) What is the current through the wire? (c) What are the voltages across the aluminum part and across the copper part?

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

(III) A 2.5-kΩ and a 3.7-kΩ resistor are connected in parallel; this combination is connected in series with a 1.4-kΩ resistor. If each resistor is rated at 1.0 W (maximum without overheating), what is the maximum voltage that can be applied across the whole network?

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

(II) Three 2.20-kΩ resistors can be connected together in four different ways, making combinations of series and/or parallel circuits. What are these four ways, and what is the net resistance in each case?

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

(III) 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?

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

(I) Three 45-Ω lightbulbs and three 65-Ω lightbulbs are connected in series.

(b) What is the total resistance if all six are wired in parallel?

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

(I) Three 45-Ω lightbulbs and three 65-Ω lightbulbs are connected in series.

(a) What is the total resistance of the circuit?

(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. (b) 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?

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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?)

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(III) 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?

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

(II) Determine

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

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

(II) Determine

(b) the voltage across each resistor, and

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

(II) Determine

(c) the current through each resistor.

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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).

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[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 ∆P ≈ ∆R dP/dR | R = R₀ .] <IMAGE>

(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. <IMAGE>

(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?

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(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?

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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.

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(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? <IMAGE>

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

(II) What is the net resistance of the circuit connected to the battery in Fig. 26–46?

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Consider two unequal resistors, of resistance R₁ and R₂, that are connected either in series or in parallel. Fill in the Table below assuming the electric potential on the low-voltage end of the combination is V_A volts and the potential at the high-voltage end of the combination is V_B volts. First draw diagrams.

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Measurements made on circuits that contain large resistances can be confusing. Consider a circuit powered by a battery E = 15.000 V with a 10.00-MΩ resistor in series with an unknown resistor R. As shown in Fig. 26–92, a particular voltmeter reads V₁ = 366 mV when connected across the 10.00 -MΩ resistor and this meter reads V₂ = 7.317 V when connected across R. Determine the value of R. [Hint: Define R_V as the voltmeter’s internal resistance.]

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