27. Resistors & DC Circuits

A resistor has a current through it of 5 A. If the EMF across the resistor is 10 V, what is the resistance of this resistor?

Two resistors are made of the same material, one twice as long as the other. If the current through the shorter resistor is 5 A, what is the current through the longer resistor if they both have the same potential difference and cross-sectional area?

The conductance G of an object is defined as the reciprocal of the resistance R; that is, G = 1/R . The unit of conductance is a mho ( = ohm⁻¹) , which is also called the siemens (S). What is the conductance (in siemens) of an object that draws 380 mA of current at 3.0 V?

(III) The filament of an incandescent lightbulb has a resistance of 12 Ω at 20°C and 140 Ω when hot.

(a) Calculate the temperature of the filament when it is hot, and take into account the change in length and area of the filament due to thermal expansion (assume tungsten for which the thermal expansion coefficient is ≈ 5.5 10⁻⁶ C°⁻¹ ).

(b) In this temperature range, what is the percentage change in resistance due to thermal expansion, and what is the percentage change in resistance due solely to the change in ρ? Use Eq. 25–5.

(III) For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a 3.60-k Ω resistor from a carbon resistor and a Nichrome wire-wound resistor connected together so the total resistance is the sum of their separate resistances. What value should each of these resistors have (at 0°C) so that the combination is temperature independent?

(II) Compute the voltage drop along an 18-m length of household no. 14 copper wire (used in 15-A circuits). The wire has diameter 1.628 mm and carries a 12-A current.

(II) Two aluminum wires have the same resistance. If one has twice the length of the other, what is the ratio of the diameter of the longer wire to the diameter of the shorter wire?

(II) Can a 1.8-mm-diameter copper wire have the same resistance as a tungsten wire of the same length? Give numerical details.

(I) What is the resistance of a 5.1-m length of copper wire 1.5 mm in diameter?

(II) A bird stands on a dc electric transmission line carrying 3800 A (Fig. 25–35). The line has 2.5 x 10⁻⁵Ω resistance per meter, and the bird’s feet are 4.0 cm apart. What is the potential difference between the bird’s feet?

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(II) A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying along the x, y, and z axes, respectively (Fig. 25–36). Determine the resistance for current that passes through the solid in

(a) the x direction (Assume the resistivity is ρ = 3.0 x 10⁻⁵ Ω•m) ,

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(II) A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying along the x, y, and z axes, respectively (Fig. 25–36). Determine the resistance for current that passes through the solid in

(b) the y direction, (Assume the resistivity is ρ = 3.0 x 10⁻⁵ Ω•m) and

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(II) A rectangular solid made of carbon has sides of lengths 1.0 cm, 2.0 cm, and 4.0 cm, lying along the x, y, and z axes, respectively (Fig. 25–36). Determine the resistance for current that passes through the solid in

(c) the z direction. (Assume the resistivity is ρ = 3.0 x 10⁻⁵ Ω•m) .

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

(II) Eight identical bulbs are connected in series across a 120-V line.

(a) What is the voltage across each bulb?

(I) Calculate the terminal voltage for a battery with an internal resistance of 0.900 Ω and an emf of 6.00 V when the battery is connected in series with

(b) a 610-Ω resistor.

(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

(a) an ammeter to read currents up to 25 A, and

(I) Calculate the terminal voltage for a battery with an internal resistance of 0.900 Ω and an emf of 6.00 V when the battery is connected in series with

(a) a 61.0-Ω resistor, and

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

(II) Ten 7.0-W Christmas tree lights are connected in series to each other and to a 120-V source. What is the resistance of each bulb?

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

(II) Eight bulbs are connected in parallel to a 120-V source by two long leads of total resistance 1.4 Ω. If 210 mA flows through each bulb, what is the resistance of each, and what fraction of the total power is wasted in the leads?

(I) What voltage will produce 0.25 A of current through a 5400-Ω resistor?

(II) (a) What is the current in the 13-Ω heating element of a 240-V clothes dryer?

(b) How much charge passes through the element in 15 min? (Assume direct current.)

(II) A 4.5-V battery is connected to a bulb whose resistance is 2.3 Ω. How many electrons leave the battery per minute?

(II) An electric device draws 5.25 A at 240 V.

(b) If the resistance of the device were reduced by 7.5%, what current would be drawn at the lower voltage?

(II) A silicon diode, whose current–voltage characteristics are given in Fig. 40–38, is connected in series with a battery and an 860-Ω resistor. What battery voltage is needed to produce a 16-mA current?

(III) The filament of an incandescent lightbulb has a resistance of 12 Ω at 20°C and 140 Ω when hot.

(b) In this temperature range, what is the percentage change in resistance due to thermal expansion, and what is the percentage change in resistance due solely to the change in ρ? Use Eq. 25–5.

Household wiring has sometimes used aluminium instead of copper.

(a) Using Table 25–1, find the ratio of the resistance of a copper wire to that of an aluminum wire of the same length and diameter.

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Household wiring has sometimes used aluminium instead of copper.

(b) Typical copper wire used for home wiring in the U.S. has a diameter of 1.63 mm. What is the resistance of 125 m of this wire?

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Household wiring has sometimes used aluminium instead of copper.

(c) What would be the resistance of the same wire if it were made of aluminum?

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A Wheatstone bridge is a type of “bridge circuit” used to make measurements of resistance. The unknown resistance to be measured, Rₓ, is placed in the circuit with accurately known resistances R₁, R₂ and R₃ (Fig. 26–73). One of these, R₃, is a variable resistor which is adjusted so that when the switch is closed momentarily, the ammeter shows zero current flow.

(a) Determine Rₓ in terms of R₁ ,R₂ and R₃.

(b) If a Wheatstone bridge is “balanced” when R₁ = 685 Ω, ,R₂ = 972Ω and R₃ = 78.6Ω, what is the value of the unknown resistance?

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The cross section of a portion of wire increases uniformly as shown in Fig. 25–38 so it has the shape of a truncated cone. The diameter at one end is a and at the other it is b, and the total length along the axis is ℓ. If the material has resistivity ρ, determine the resistance R between the two ends in terms of a, b, ℓ, and ρ. Assume that the current flows uniformly through each section, and that the taper is small, i.e., ( b -a) ≪ ℓ.

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Electricity can be a hazard in hospitals, particularly to patients who are connected to electrodes, such as an ECG. Suppose that the motor of a motorized bed shorts out to the bed frame, and the bed frame’s connection to a ground has broken (or was not there in the first place). If a nurse touches the bed and the patient at the same time, the nurse becomes a conductor and a complete circuit can be made through the patient to ground through the ECG apparatus. This is shown schematically in Fig. 26–79. Calculate the current through the patient.

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A sequence of potential differences V is applied across a wire ( diameter = 0.32 mm, length = 11cm) and the resulting currents I are measured as follows:

(b) Plot I vs. V. Based on this plot, can you conclude that the wire obeys Ohm’s law (i.e., did you obtain a straight line with the expected y-intercept, within the values of the significant figures)? If so, determine the wire’s resistance R.

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A zener diode voltage regulator is shown in Fig. 40–55. Suppose that R = 2.80 kΩ and that the diode breaks down at a reverse voltage of 130 V. (The current increases rapidly at this point, as shown on the far left of Fig. 40–38 at a voltage of -12V on that diagram.) The diode is rated at a maximum current of 120 mA. (a) If Rload = 21.0 kΩ, over what range of supply voltages will the circuit maintain the output voltage at 130 V? (b) If the supply voltage is 275 V, over what range of load resistance will the voltage be regulated?

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