Cleophas Rwemera: February 2011

Chapter 31

Skills Builders problems

Introduction to Electric Current

Learning Goal: To understand the nature of electric current and the conditions under which it exists.

Electric current is defined as the motion of electric charge through a conductor. Conductors are materials that contain movable charged particles. In metals, the most commonly used conductors, such charged particles are electrons. The more electrons that pass through a cross section of a conductor per second, the greater the current. The conventional definition of current is

$I=\frac{Q_{\rm total}}{\Delta t}$ ,

where

is the current in a conductor and

is the total charge passing through a cross section of the conductor during the time interval

The motion of free electrons in metals not subjected to an electric field is random: Even though the electrons move fairly rapidly, the net result of such motion is that $Q_{\rm total}=0$ (i.e., equal numbers of electrons pass through the cross section in opposite directions). However, when an electric field is imposed, the electrons continue in their random motion, but in addition, they tend to move in the direction of the force applied by the electric field.

In summary, the two conditions for electric current in a material are the presence of movable charged particles in the material and the presence of an electric field.

Quantitatively, the motion of electrons under the influence of an electric field is described by the drift speed, which tends to be much smaller than the speed of the random motion of the electrons. The number of electrons passing through a cross section of a conductor depends on the drift speed (which, in turn, is determined by both the microscopic structure of the material and the electric field) and the cross-sectional area of the conductor.

In this problem, you will be offered several conceptual questions that will help you gain an understanding of electric current in metals.

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Part A

You are presented with several long cylinders made of different materials. Which of them are likely to be good conductors of electric current?

Check all that apply.

	copper
	aluminum
	glass
	quartz
	cork
	plywood
	table salt
	gold

Correct

Part B

Metals are good conductors of electric current for which of the following reasons?

	They possess high concentrations of protons.
	They possess low concentrations of protons.
	They possess high concentrations of free electrons.
	They possess low concentrations of free electrons.

Correct

Part C

Which of the following is the most likely drift speed of the electrons in the filament of a light bulb?

Correct

Part D

You are presented with several wires made of the same conducting material. The radius and drift speed are given for each wire in terms of some unknown units $\rm r$ and $\rm v$ . Rank the wires in order of decreasing electron current.

Rank from most to least electron current. To rank items as equivalent, overlap them.

Correct

Part E

The drift speed of the electrons in a wire depends strongly on which of the following factors?

	the cross-sectional area of the wire
	the mass of the wire
	the temperature of the wire
	the internal electric field in the wire

Correct

Part F

What quality must the charge density on the surface of a conducting wire possess if an electric field is to act on the negatively charged electrons inside the wire?

The charge density must be

	positive.
	negative.
	nonuniform.
	uniform.

Correct

Current and Current Density at a Junction

Consider the juncion of three wires as shown in the diagram.

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The magnitudes of the current density and the diameters for wires 1 and 2 are given in the table. The current directions are indicated by the arrows.

Wire	Current density ( $\rm A/mm^2$ )	Diameter ( $\rm mm$ )
1	3.0	2.0
2	5.0	3.0

Part A

Find the current

in wire 3.

Express your answer in amperes to two significant figures. Call current out of the junction positive and current into the junction negative.

=-26 $\rm A$

Correct

Part B

Find the magnitude of the current density

in wire 3. The diameter of wire 3 is 1.5 millimeters.

Express your answer in amperes per square millimeter to two significant figures.

=15 $\rm A/mm^2$ .

Correct

Same Value Problems

Problem 31.6

The electron drift speed is $2.0 \times 10^{ - 4}\;{\rm m}/{\rm s}$ in a metal with a mean free time between collisions of $5.0 \times 10^{ - 14}\;{\rm s}$ .

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Part A

What is the electric field strength?

2.28×10⁻² N/C

Correct

Problem 31.11

The wires leading to and from a 0.12- ${\rm mm}$ -diameter lightbulb filament are 1.5 ${\rm mm}$ in diameter. The wire to the filament carries a current with a current density of $4.5\times10^5\;{\rm A}/{\rm m}^{2}$ .

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Part A

What is the current in the filament?

Express your answer using two significant figures.

=0.80 ${\rm A}$

Correct

Part B

What is the current density in the filament?

Express your answer using two significant figures.

=7.0×10⁷ ${\rm A/m^2}$

Correct

Problem 31.23

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Part A

What electric field strength is needed to create a 5.0 $\rm A$ current in a 2.0- $\rm mm$ -diameter iron wire?

0.159 $\rm N/C$

Correct

Problem 31.30

Wires 1 and 2 are made of the same metal. Wire 2 has twice the length and twice the diameter of wire 1. What are the ratios (a) $\rho _{2} / \rho _{1}$ of the resistivities and (b) $R_{2} / R_{1}$ of the resistances of the two wires?