Problem 59
Show that the power supplied to the circuit in the figure by the battery with internal resistance is maximum when the resistance of the resistor in the circuit, \(R\), is equal to \(R_{i}\). Determine the power supplied to \(R\). For practice, calculate the power dissipated by a \(12.0-\mathrm{V}\) battery with an internal resistance of \(2.00 \Omega\) when \(R=1.00 \Omega, R=2.00 \Omega,\) and \(R=3.00 \Omega\)
Problem 60
A water heater consisting of a metal coil that is connected across the terminals of a 15 -V power supply is able to heat \(250 \mathrm{~mL}\) of water from room temperature to boiling point in \(45 \mathrm{~s}\). What is the resistance of the coil?
Problem 61
A potential difference of \(V=0.500 \mathrm{~V}\) is applied across a block of silicon with resistivity \(8.70 \cdot 10^{-4} \Omega \mathrm{m}\). As indicated in the figure, the dimensions of the silicon block are width \(a=2.00 \mathrm{~mm}\) and length \(L=15.0 \mathrm{~cm} .\) The resistance of the silicon block is \(50.0 \Omega\), and the density of charge carriers is \(1.23 \cdot 10^{23} \mathrm{~m}^{-3}\) Assume that the current density in the block is uniform and that current flows in silicon according to Ohm's Law. The total length of 0.500 -mm-diameter copper wire in the circuit is \(75.0 \mathrm{~cm},\) and the resistivity of copper is \(1.69 \cdot 10^{-8} \Omega \mathrm{m}\) a) What is the resistance, \(R_{w}\) of the copper wire? b) What are the direction and the magnitude of the electric current, \(i\), in the block? c) What is the thickness, \(b\), of the block? d) On average, how long does it take an electron to pass from one end of the block to the other? \(?\) e) How much power, \(P\), is dissipated by the block? f) In what form of energy does this dissipated power appear?
Problem 62
In an emergency, you need to run a radio that uses \(30.0 \mathrm{~W}\) of power when attached to a \(10.0-\mathrm{V}\) power supply. The only power supply you have access to provides \(25.0 \mathrm{kV}\) but you do have a large number of \(25.0-\Omega\) resistors. If you want the power to the radio to be as close as possible to \(30.0 \mathrm{~W}\), how many resistors should you use, and how should they be connected (in series or in parallel)?
Problem 63
A certain brand of hot dog cooker applies a potential difference of \(120 \mathrm{~V}\) to opposite ends of the hot dog and cooks it by means of the heat produced. If \(48 \mathrm{~kJ}\) is needed to cook each hot dog, what current is needed to cook three hot dogs simultaneously in \(2.0 \mathrm{~min}\) ? Assume a parallel connection.
Problem 65
The resistivity of a conductor is \(\rho=1.00 \cdot 10^{-5} \Omega \mathrm{m}\). If a cylindrical wire is made of this conductor, with a crosssectional area of \(1.00 \cdot 10^{-6} \mathrm{~m}^{2},\) what should the length of the wire be for its resistance to be \(10.0 \Omega ?\)
Problem 66
Two cylindrical wires of identical length are made of copper and aluminum. If they carry the same current and have the same potential difference across their length, what is the ratio of their radii?
Problem 67
Two resistors with resistances \(200 . \Omega\) and \(400 . \Omega\) are connected (a) in series and (b) in parallel with an ideal 9.00-V battery. Compare the power delivered to the \(200 .-\Omega\) resistor.
Problem 68
What is (a) the conductance and (b) the radius of a \(3.5-\mathrm{m}\) -long iron heating element for a \(110-\mathrm{V}, 1500-\mathrm{W}\) heater?
Problem 70
A modern house is wired for \(115 \mathrm{~V}\), and the current is limited by circuit breakers to a maximum of \(200 .\) A. (For the purpose of this problem, treat these as DC quantities.) a) Calculate the minimum total resistance the circuitry in the house can have at any time. b) Calculate the maximum electrical power the house can consume.
