Current and Resistance

The current through the battery and resistors 1 and 2 in Figure-a is 2.00 A. Energy is transferred from the current to thermal energy E_th in both resistors. Curves 1 and 2 in Figure-b give that thermal energy E_th for resistors 1 and 2, respectively, as a function of time t. The vertical scale is set by ${\mathit{E}}_{\mathbf{t}\mathbf{h}\mathbf{,}\mathbf{s}}\mathbf{=}\mathbf{40}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mJ}$, and the horizontal scale is set by ${\mathit{t}}_{\mathbf{s}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{s}$. What is the power of the battery?

Figure 26-15 shows cross sections through three long conductors of the same length and material, with square cross sections of edge lengths as shown. Conductor B fits snugly within conductor A, and conductor C fits snugly within conductor B. Rank the following according to their end-to-end resistances, greatest first: the individual conductors and the combinations of A +B (B inside A),B+C (C inside B), and A +B +C (B inside A inside C).

Figure 26-16 shows cross sections through three wires of identical length and material; the sides are given in millimeters. Rank the wires according to their resistance (measured end to end along each wire’s length), greatest first.

Figure 26-17 shows a rectangular solid conductor of edge lengths L, 2L, and 3L. A potential difference V is to be applied uniformly between pairs of opposite faces of the conductor as in Fig. 26-8b. (The potential difference is applied between the entire face on one side and the entire face on the other side.) First V is applied between the left–right faces, then between the top–bottom faces, and then between the front–back faces. Rank those pairs, greatest first, according to the following (within the conductor): (a) the magnitude of the electric field, (b) the current density, (c) the current, and (d) the drift speed of the electrons.

Figure 26-18, shows plots of the current I through a certain cross section of a wire over four different time periods. Rank the periods according to the net charge that passes through the cross section during the period, greatest first.

During the 4.0 min a 5.0 A current is set up in a wire, how many  (a) coulombs pass through any cross section across the wire’s width?  (b) electrons pass through any cross section across the wire’s width?

An isolated conducting sphere has a 10 cm radius. One wire carries a current of 1.000 0020A into it. Another wire carries a current of out of it. How long would it take for the sphere to increase in potential by 1000V ?

A charged belt, 50 cm wide, travels at 30m/s between a source of charge and a sphere. The belt carries charge into the sphere at a rate corresponding to $\mathbf{100}\mathbf{\mu A}$ . Compute the surface charge density on the belt.

The (United States) National Electric Code, which sets maximum safe currents for insulated copper wires of various diameters, is given (in part) in the table. Plot the safe current density as a function of diameter. Which wire gauge has the maximum safe current density? (“Gauge” is a way of identifying wire diameters, and$\mathbf{1}\mathbf{mil}\mathbf{=}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$in.)

Figure 26-19 shows four situations in which positive and negative charges move horizontally and gives the rate at which each charge moves. Rank the situations according to the effective current through the regions, greatest first.

A beam contains $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}$ doubly charged positive ions per cubic centimetre , all of which are moving north with a speed of $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}\mathbf{ }}\mathit{m}\mathbf{/}\mathit{s}$. What is the (a) magnitude of the current density $\overrightarrow{\mathbf{J}}$ ? (b) Direction of the current density $\overrightarrow{\mathbf{J}}$? (c) What additional quantity do you need to calculate the total current i of this ion beam?

A certain cylindrical wire carries current. We draw a circle of radius r around its central axis in figure-a to determine the current i within the circle. Figure-b shows current i as a function of r². The vertical scale is set by ${\mathit{i}}_{\mathbf{s}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{m}\mathit{A}$ ,and the horizontal scale is set by, ${\mathit{r}}_{\mathbf{s}}^{\mathbf{2}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}{\mathit{m}}^{\mathbf{2}}$ . (a) Is the current density uniform?  (b) If so, what is its magnitude?

In Fig. 26-20, a wire that carries a current consists of three sections with different radii. Rank the sections according to the following quantities, greatest first: (a) current, (b) magnitude of current density, and (c) magnitude of electric field.

Figure 26-21 gives the electric potential V(x) versus position xalong a copper wire carrying current.The wire consists of three sections that differ in radius. Rank the three sections according to the magnitude of the (a) electric field and (b) current density, greatest first.

The following table gives the lengths of three copper rods, their diameters, and the potential differences between their ends. Rank the rods according to (a) the magnitude of the electric field within them, (b) the current density within them, and (c) the drift speed of electrons through them, greatest first.

Figure 26-22 gives the drift speed v_d of conduction electrons in a copper wire versus position x along the wire. The wire consists of three sections that differ in radius. Rank the three sections according to the following quantities, greatest first: (a) radius, (b) number of conduction electrons per cubic meter, (c) magnitude of electric field, (d) conductivity

Three wires, of the same diameter, are connected in turn between two points maintained at a constant potential difference. Their resistivity and lengths are p and L (wire A),  1.2p and 1.2 L (wire B), and 0.9p and L (wire C). Rank the wire according to the rate at which energy is transferred to thermal energy within them, greatest first.

Figure 26-23 gives, for three wires of radius R, the current density J(r) versus radius r, as measured from the center of a circular cross section through the wire. The wires are all made from the same material. Rank the wires according to the magnitude of the electric field (a) at the center, (b) halfway to the surface, and (c) at the surface, greatest first.

A fuse in an electric circuit is a wire that is designed to melt, and thereby open the circuit, if the current exceeds a predetermined value. Suppose that the material to be used in a fuse melts when the current density rises to $\mathbf{440}\mathbf{ }\mathit{A}\mathbf{/}\mathit{c}{\mathit{m}}^{\mathbf{2}}$. What diameter of cylindrical wire should be used to make a fuse that will limit the current to 0.50 A?

A small but measurable current of $\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{10}}\mathit{A}$ exists in a copper wire whose diameter is 2.5 mm .The number of charge carriers per unit volume is $\mathbf{8}\mathbf{.}\mathbf{49}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}{\mathit{m}}^{\mathbf{-}\mathbf{3}}$ . Assuming the current is uniform, calculate the (a) current density and (b) electron drift speed.

The magnitude J(r) of the current density in a certain cylindrical wire is given as a function of radial distance from the centre of the wire’s cross section as J(r) = Br, where r is in meters, J is in amperes per square meter, and $\mathit{B}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}\mathbf{ }}\mathit{A}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$ .This function applies out to the wire’s radius of 2.00 mm . How much current is contained within the width of a thin ring concentric with the wire if the ring has a radial width of $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{\mu m}$ and is at a radial distance of 1.20 mm ?

The magnitude J of the current density in a certain lab wire with a circular cross section of radius R = 2.00 mm is given by $\mathit{J}\mathbf{=}\left(3.00\times {10}^8\right){\mathit{r}}^{\mathbf{2}}$ , with J in amperes per square meter and radial distance r in meters. What is the current through the outer section bounded by r = 0.900 R and r = R ?

What is the current in a wire of radius R=3.40 mm if the magnitude of the current density is given by (a)${\mathbf{J}}_{\mathbf{a}}\mathbf{=}{\mathbf{J}}_{\mathbf{0}}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{ }{\mathbf{J}}_{\mathbf{b}}\mathbf{=}{\mathbf{J}}_{\mathbf{0}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}$, in which r is the radial distance and ${\mathbf{J}}_{\mathbf{0}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{ }\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$?  (c) Which function maximizes the current density near the wire’s surface?

Near Earth, the density of protons in the solar wind (a stream of particles from the Sun) is $\mathbf{8}\mathbf{.}\mathbf{70}\mathbf{ }\mathit{c}{\mathit{m}}^{\mathbf{-}\mathbf{3}}$ , and their speed is 470 km/s .  (a) Find the current density of these protons. (b) If Earth’s magnetic field did not deflect the protons, what total current would Earth receive?

How long does it take electrons to get from a car battery to the starting motor? Assume the current is 300 A and the electrons travel through a copper wire with cross-sectional area $\mathbf{0}\mathbf{.}\mathbf{21}\mathit{c}{\mathit{m}}^{\mathbf{2}}$ and length 0.85 m . The number of charge carriers per unit volume is $\mathbf{8}\mathbf{.}\mathbf{49}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}{\mathit{m}}^{\mathbf{-}\mathbf{3}}$ .

A human being can be electrocuted if a current as small as 50mA passes near the heart. An electrician working with sweaty hands makes good contact with the two conductors he is holding, one in each hand. If his resistance is $\mathbf{2000}\mathbf{\Omega }$, what might the fatal voltage be?

A coil is formed by winding 250turns of insulated 16-gauge copper wire (diameter = 1.3 mm) in a single layer on a cylindrical form of radius 12cm. What is the resistance of the coil? Neglect the thickness of the insulation. (Use Table 26-1.)

Copper and aluminum are being considered for a high-voltage transmission line that must carry a current of 60.0 A. The resistance per unit length is to be $\mathbf{0}\mathbf{.}\mathbf{150}\mathbf{ }\mathbf{\Omega }\mathbf{/}\mathbf{km}$. The densities of copper and aluminum are 8960 and $\mathbf{2600}\mathbf{ }\mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$, respectively. Compute (a) the magnitude J of the current density and (b) the mass per unit length $\mathit{\lambda }$ for a copper cable and (c) J for an aluminum cable (d)  for an aluminum cable.

A wire of Nichrome (a nickel–chromium–iron alloy commonly used in heating elements) is 1.0m long and $\mathbf{1}\mathbf{.}\mathbf{0}{\mathbf{mm}}^{\mathbf{2}}$ in cross-sectional area. It carries a current of 4.0 A when a 2.0 V potential difference is applied between its ends. Calculate the conductivity  of Nichrome.

A wire 4.00m long and 6.00mm in diameter has a resistance of $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{m\Omega }$. A potential difference of 23.0 V is applied between the ends. (a) What is the current in the wire?   (b) What is the magnitude of the current density?  (c) Calculate the resistivity of the wire material.  (d) Using Table, identify the material.

What is the resistivity of a wire of diameter 1.0 mm, length 2.0 m, and $\mathbf{50}\mathbf{m\Omega }$ resistance?

A certain wire has a resistance R. What is the resistance of a second wire, made of the same material, that is half as long and has half the diameter?

A common flashlight bulb is rated at 0.30 A and 2.9 V (the values of the current and voltage under operating conditions). If the resistance of the tungsten bulb filament at room temperature ($\mathbf{20}\mathbf{°}\mathbf{C}$) is $\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\Omega }$, what is the temperature of the filament when the bulb is on?

Kiting during a storm. The legend that Benjamin Franklin flew a kite as a storm approached is only a legend—he was neither stupid nor suicidal. Suppose a kite string of radius 2.00 mm extends directly upward by 0.800 km and is coated with a 0.500 mm layer of water having resistivity $\mathbf{150}\mathbf{\Omega m}$. If the potential difference between the two ends of the string is 160 MV, what is the current through the water layer? The danger is not this current but the chance that the string draws a lightning strike, which can have a current as large as 500 000 A (way beyond just being lethal).

When 115 V is applied across a wire that is 10 m long and has a 0.33mm radius, the magnitude of the current density is $\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$ . Find the resistivity of the wire.

Figure-a gives the magnitude E(x) of the electric fields that have been set up by a battery along a resistive rod of length 9.00mm (Figure-b). The vertical scale is set by. ${\mathbf{E}}_{\mathbf{s}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{V}\mathbf{/}\mathbf{m}$ The rod consists of three sections of the same material but with different radii. (The schematic diagram of Figure-b does not indicate the different radii.) The radius of section 3 is 2.00mm. (a) What is the radius of section 1 and (b) What is the radius of section 2? 

A wire with a resistance of 6.0$\mathbf{\Omega }$ is drawn out through a die so that its new length is three times its original length. Find the resistance of the longer wire, assuming that the resistivity and density of the material are unchanged.

In Figure-a, a $\mathbf{9}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{V}$battery is connected to a resistive strip that consists of three sections with the same cross-sectional areas but different conductivities. Figure-b gives the electric potential V(x) versus position x along the strip. The horizontal scale is set by ${\mathbf{x}}_{\mathbf{s}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{mm}$. Section 3 has conductivity $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{ }{\left(\mathrm{\Omega m}\right)}^{\mathbf{-}\mathbf{1}}$. (a) What is the conductivity of section 1 and (b) What is the conductivity of section 2?

Two conductors are made of the same material and have the same length. Conductor A is a solid wire of diameter . Conductor B is a hollow tube of outside diameter 1.0 mm and inside diameter 1.0mm. What is the resistance ratio ${\mathit{R}}_{\mathit{A}}\mathit{/}{\mathit{R}}_{\mathit{B}}$, measured between their ends?

Figure gives the electric potential V(x) along a copper wire carrying uniform current, from a point of higher potential ${\mathbf{V}}_{\mathbf{s}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{\mu V}$ at x = 0  to a point of zero potential at ${\mathbf{x}}_{\mathbf{s}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{m}$ . The wire has a radius of  . What is the current in the wire?

A potential difference of 3.00nV is set up across a 2.00cm length of copper wire that has a radius of 2.00mm. How much charge drifts through a cross-section in 3.00 ms?

If the gauge number of a wire is increased by 6, the diameter is halved; if a gauge number is increased by 1, the diameter decreases by the factor 2^1/6 (see the table). Knowing this, and knowing that of 10-gauge copper wire has a resistance of approximately, $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{\Omega }$, estimate the resistance of 25 ft of 22-gauge copper wire.

An electrical cable consists of 125 strands of fine wire, each having $\mathbf{2}\mathbf{.}\mathbf{65}\mathbf{ }\mathit{\mu }\mathit{\Omega }$ resistance. The same potential difference is applied between the ends of all the strands and results in a total current of 0.750 A .  (a) What is the current in each strand?  (b) What is the applied potential difference?  (c) What is the resistance of the cable?

Earth’s lower atmosphere contains negative and positive ions that are produced by radioactive elements in the soil and cosmic rays from space. In a certain region, the atmospheric electric field strength is 120V/m and the field is directed vertically down. This field causes singly charged positive ions, at a density of 620cm^-3 , to drift downward and singly charged negative ions, at a density of 550cm^-3 , to drift upward (Figure). The measured conductivity of the air in that region is $\mathbf{2}\mathbf{.}\mathbf{70}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{14}}{\left(\Omega ·m\right)}^{\mathbf{-}\mathbf{1}}$ . Calculate (a) the magnitude of the current density and (b) the ion drift speed, assumed to be the same for positive and negative ions.

A block in the shape of a rectangular solid has a cross-sectional area of 3.50cm² across its width, a front-to-rear length of 15.8cm , and a resistance of $\mathbf{935}\mathbf{\Omega }$. The block’s material contains  $\mathbf{5}\mathbf{.}\mathbf{33}\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}$ conduction electrons/m³. A potential difference of 35.8 V is maintained between its front and rear faces.  (a) What is the current in the block? (b) If the current density is uniform, what is its magnitude? What are (c) the drift velocity of the conduction electrons and (d) the magnitude of the electric field in the block?

Figure shows wire section 1 of diameter ${\mathit{D}}_{\mathbf{1}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathit{R}$ and wire section 2 of diameter ${\mathit{D}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{R}$, connected by a tapered section. The wire is copper and carries a current. Assume that the current is uniformly distributed across any cross-sectional area through the wire’s width. The electric potential change V along the length L=2.00m  shown in section 2 is $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathit{V}\mathbf{ }$. The number of charge carriers per unit volume is $\mathbf{8}\mathbf{.}\mathbf{49}\mathbf{\times }{\mathbf{10}}^{\mathbf{2}}\mathbf{8}{\mathbf{m}}^{\mathbf{-}\mathbf{3}}$. What is the drift speed of the conduction electrons in section 1?

In Figure, current is set up through a truncated right circular cone of resistivity $\mathbf{731}\mathbf{\Omega }\mathbf{.}\mathbf{m}$ , left radius a = 2.00mm, right radius b = 2.30mm, and length L = 1.94 cm. Assume that the current density is uniform across any cross section taken perpendicular to the length. What is the resistance of the cone?

Swimming during a storm. Figure shows a swimmer at distance D = 35.0m from a lightning strike to the water, with current l = 78 kA. The water has resistivity $\mathbf{30}\mathbf{\Omega m}$ , the width of the swimmer along a radial line from the strike is 0.70m , and his resistance across that width is $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{k\Omega }$ . Assume that the current spreads through the water over a hemisphere centered on the strike point. What is the current through the swimmer?

Show that, according to the free-electron model of electrical conduction in metals and classical physics, the resistivity of metals should be proportional to $\sqrt{\mathbf{T}}$ where T is the temperature in kelvins.

In Figure-a, a $\mathbf{20}\mathit{\Omega }$ resistor is connected to a battery. Figure-b shows the increase of thermal energy ${\mathit{E}}_{\mathbf{t}\mathbf{h}}$ in the resistor as a function of time t. The vertical scale is set by ${\mathit{E}}_{\mathbf{th}\mathbf{,}\mathbf{s}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{50}\mathit{m}\mathit{J}$, and the horizontal scale is set by  ${\mathit{t}}_{\mathbf{s}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$. What is the electric potential across the battery?