Problem 28
A copper wire has a diameter \(d_{\mathrm{Cu}}=0.0500 \mathrm{~cm}\) is \(3.00 \mathrm{~m}\) long, and has a density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). As shown in the figure, the copper wire is attached to an equal length of aluminum wire with a diameter \(d_{\mathrm{A} \mathrm{I}}=0.0100 \mathrm{~cm}\) and density of charge carriers of \(6.02 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). A current of 0.400 A flows through the copper wire. a) What is the ratio of the current densities in the two wires, \(J_{\mathrm{Cu}} / J_{\mathrm{Al}} ?\) b) What is the ratio of the drift velocities in the two wires, \(v_{\mathrm{d}-\mathrm{Cu}} / v_{\mathrm{d}-\mathrm{Al}} ?\)
Problem 29
A current of \(0.123 \mathrm{~mA}\) flows in a silver wire whose cross-sectional area is \(0.923 \mathrm{~mm}^{2}\) a) Find the density of electrons in the wire, assuming that there is one conduction electron per silver atom. b) Find the current density in the wire assuming that the current is uniform. c) Find the electron's drift speed.
Problem 30
What is the resistance of a copper wire of length \(l=\) \(10.9 \mathrm{~m}\) and diameter \(d=1.3 \mathrm{~mm} ?\) The resistivity of copper is \(1.72 \cdot 10^{-8} \Omega \mathrm{m}\)
Problem 31
Two conductors are made of the same material and have the same length \(L\). Conductor \(\mathrm{A}\) is a hollow tube with inside diameter \(2.00 \mathrm{~mm}\) and outside diameter \(3.00 \mathrm{~mm} ;\) conductor \(\mathrm{B}\) is a solid wire with radius \(R_{\mathrm{B}}\). What value of \(R_{\mathrm{B}}\) is required for the two conductors to have the same resistance measured between their ends?
Problem 34
A rectangular wafer of pure silicon, with resistivity \(\rho=2300 \Omega \mathrm{m},\) measures \(2.00 \mathrm{~cm}\) by \(3.00 \mathrm{~cm}\) by \(0.010 \mathrm{~cm}\) Find the maximum resistance of this rectangular wafer between any two faces.
Problem 35
A copper wire that is \(1 \mathrm{~m}\) long and has a radius of \(0.5 \mathrm{~mm}\) is stretched to a length of \(2 \mathrm{~m}\). What is the fractional change in resistance, \(\Delta R / R,\) as the wire is stretched? What is \(\Delta R / R\) for a wire of the same initial dimensions made out of aluminum?
Problem 36
The most common material used for sandpaper, silicon carbide, is also widely used in electrical applications. One common device is a tubular resistor made of a special grade of silicon carbide called carborundum. A particular carborundum resistor (see the figure) consists of a thick-walled cylindrical shell (a pipe) of inner radius \(a=\) \(1.50 \mathrm{~cm},\) outer radius \(b=2.50 \mathrm{~cm},\) and length \(L=60.0 \mathrm{~cm} .\) The resistance of this carborundum resistor at \(20 .{ }^{\circ} \mathrm{C}\) is \(1.00 \Omega\). a) Calculate the resistivity of carborundum at room temperature. Compare this to the resistivities of the most commonly used conductors (copper, aluminum, and silver). b) Carborundum has a high temperature coefficient of resistivity: \(\alpha=2.14 \cdot 10^{-3} \mathrm{~K}^{-1} .\) If, in a particular application, the carborundum resistor heats up to \(300 .{ }^{\circ} \mathrm{C},\) what is the percentage change in its resistance between room temperature \(\left(20 .{ }^{\circ} \mathrm{C}\right)\) and this operating temperature?
Problem 38
A potential difference of \(12.0 \mathrm{~V}\) is applied across a wire of cross-sectional area \(4.50 \mathrm{~mm}^{2}\) and length \(1000 . \mathrm{km} .\) The current passing through the wire is \(3.20 \cdot 10^{-3} \mathrm{~A}\). a) What is the resistance of the wire? b) What type of wire is this?
Problem 39
One brand of \(12.0-\mathrm{V}\) automotive battery used to be advertised as providing " 600 cold-cranking amps." Assuming that this is the current the battery supplies if its terminals are shorted, that is, connected to negligible resistance, determine the internal resistance of the battery.
Problem 40
A copper wire has radius \(r=0.0250 \mathrm{~cm},\) is \(3.00 \mathrm{~m}\) long, has resistivity \(\rho=1.72 \cdot 10^{-8} \Omega \mathrm{m},\) and carries a current of \(0.400 \mathrm{~A}\). The wire has density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\) a) What is the resistance, \(R,\) of the wire? b) What is the electric potential difference, \(\Delta V\), across the wire? c) What is the electric field, \(E\), in the wire?
