Current

Chapter contents: Chapter 9: Electricity

Electric Current

Electric current is defined to be the rate at which charge flows. A large current, such as that used to start a truck engine, moves a large amount of charge in a small time, whereas a small current, such as that used to operate a hand-held calculator, moves a small amount of charge over a long period of time. In equation form, electric current

I

is defined to be

I = \frac{Δ Q}{Δ t},

(9.8.1)

where

Δ Q

is the amount of charge passing through a given area in time

Δ t

. (As in previous chapters, initial time is often taken to be zero, in which case

Δ t = t

.) (See Figure 9.8.1.) The SI unit for current is the ampere (A), named for the French physicist André-Marie Ampère (1775–1836). Since

I = Δ Q / Δ t

, we see that an ampere is one coulomb per second:

1 A = 1 C/s

(9.8.2)

Not only are fuses and circuit breakers rated in amperes (or amps), so are many electrical appliances.

Charges are shown as small spheres moving through a section of a conducting wire. The direction of movement of charge is indicated by arrows along the length of the conductor toward the right. The cross-sectional area of the wire is labeled as A. The current is equal to the flow of charge. — **Figure 9.8.1.** The rate of flow of charge is current. An ampere is the flow of one coulomb through an area in one second.

Example 9.8.1

Calculating Currents: Current in a Truck Battery and a Handheld Calculator

(a) What is the current involved when a truck battery sets in motion 720 C of charge in 4.00 s while starting an engine? (b) How long does it take 1.00 C of charge to flow through a handheld calculator if a 0.300-mA current is flowing?

Strategy

We can use the definition of current in the equation

I = Δ Q / Δ t

to find the current in part (a), since charge and time are given. In part (b), we rearrange the definition of current and use the given values of charge and current to find the time required.

Solution for (a)

Entering the given values for charge and time into the definition of current gives

\begin{array}{l} I & = & \frac{Δ Q}{Δ t} = \frac{720 C}{4.00 s} = 180 C/s \\ = & 180 A. \end{array}

(9.8.3)

Discussion for (a)

This large value for current illustrates the fact that a large charge is moved in a small amount of time. The currents in these “starter motors” are fairly large because large frictional forces need to be overcome when setting something in motion.

Solution for (b)

Solving the relationship

I = Δ Q / Δ t

for time

Δ t

, and entering the known values for charge and current gives

\begin{array}{l} Δ t & = & \frac{Δ Q}{I} = \frac{1.00 C}{0.300 \times 10^{- 3} C/s} \\ = & 3.33 \times 10^{3} s. \end{array}

(9.8.4)

Discussion for (b)

This time is slightly less than an hour. The small current used by the hand-held calculator takes a much longer time to move a smaller charge than the large current of the truck starter. So why can we operate our calculators only seconds after turning them on? It’s because calculators require very little energy. Such small current and energy demands allow handheld calculators to operate from solar cells or to get many hours of use out of small batteries. Remember, calculators do not have moving parts in the same way that a truck engine has with cylinders and pistons, so the technology requires smaller currents.

Figure 9.8.2 shows a simple circuit and the standard schematic representation of a battery, conducting path, and load (a resistor). Schematics are very useful in visualizing the main features of a circuit. A single schematic can represent a wide variety of situations. The schematic in Figure 9.8.2 (b), for example, can represent anything from a truck battery connected to a headlight lighting the street in front of the truck to a small battery connected to a penlight lighting a keyhole in a door. Such schematics are useful because the analysis is the same for a wide variety of situations. We need to understand a few schematics to apply the concepts and analysis to many more situations.

Part a shows a bulb glowing when its terminals are connected to a battery through a wire. The voltage of the battery is labeled as V. The current through the bulb is represented as I, and the current direction is shown using arrows emerging from the positive terminal of the battery, passing through the bulb, and entering the negative terminal of the battery. Part b shows an electric circuit diagram with a resistance connected across the terminals of a battery of voltage V. The current is shown using arrows as emerging from the positive terminal of the battery, passing through the resistance, and entering the negative terminal of the battery. — **Figure 9.8.2.** (a) A simple electric circuit. A closed path for current to flow through is supplied by conducting wires connecting a load to the terminals of a battery. (b) In this schematic, the battery is represented by the two parallel red lines, conducting wires are shown as straight lines, and the zigzag represents the load. The schematic represents a wide variety of similar circuits.

Note that the direction of current flow in Figure 9.8.2 is from positive to negative. The direction of conventional current is the direction that positive charge would flow. Depending on the situation, positive charges, negative charges, or both may move. In metal wires, for example, current is carried by electrons—that is, negative charges move. In ionic solutions, such as salt water, both positive and negative charges move. This is also true in nerve cells. A Van de Graaff generator used for nuclear research can produce a current of pure positive charges, such as protons. Figure 9.8.3 illustrates the movement of charged particles that compose a current. The fact that conventional current is taken to be in the direction that positive charge would flow can be traced back to American politician and scientist Benjamin Franklin in the 1700s. He named the type of charge associated with electrons negative, long before they were known to carry current in so many situations. Franklin, in fact, was totally unaware of the small-scale structure of electricity.

It is important to realize that there is an electric field in conductors responsible for producing the current, as illustrated in Figure 9.8.3. Unlike static electricity, where a conductor in equilibrium cannot have an electric field in it, conductors carrying a current have an electric field and are not in static equilibrium. An electric field is needed to supply energy to move the charges.

Making Connections: Take-Home Investigation—Electric Current Illustration

Find a straw and little peas that can move freely in the straw. Place the straw flat on a table and fill the straw with peas. When you pop one pea in at one end, a different pea should pop out the other end. This demonstration is an analogy for an electric current. Identify what compares to the electrons and what compares to the supply of energy. What other analogies can you find for an electric current?

Note that the flow of peas is based on the peas physically bumping into each other; electrons flow due to mutually repulsive electrostatic forces.

In part a, positive charges move toward the right through a conducting wire. The direction of movement of charge is indicated by arrows along the length of the wire. The area of a cross section of the wire is labeled as A. The direction of the electric field E is toward the right, in the same direction as movement of positive charge. The current direction is also toward the right, shown by an arrow. In part b, negative charges move toward the left through a conducting wire. The direction of movement of charge is indicated by arrows along the length of the wire. The area of a cross section of the wire is labeled as A. The direction of the electric field E is toward the right, opposite the direction of movement of negative charge. The current direction is also toward the right, shown by an arrow. — **Figure 9.8.3.** Current $I$ is the rate at which charge moves through an area $A$ , such as the cross-section of a wire. Conventional current is defined to move in the direction of the electric field. (a) Positive charges move in the direction of the electric field and the same direction as conventional current. (b) Negative charges move in the direction opposite to the electric field. Conventional current is in the direction opposite to the movement of negative charge. The flow of electrons is sometimes referred to as electronic flow.

Example 9.8.2

Calculating the Number of Electrons that Move through a Calculator

If the 0.300-mA current through the calculator mentioned in the Example 9.8.1 example is carried by electrons, how many electrons per second pass through it?

Strategy

The current calculated in the previous example was defined for the flow of positive charge. For electrons, the magnitude is the same, but the sign is opposite,

I_{electrons} = - 0.300 \times 10^{-3} C/s

.Since each electron

(e^{-})

has a charge of

–1 . 60 \times 10^{- 19} C

, we can convert the current in coulombs per second to electrons per second.

Solution

Starting with the definition of current, we have

I_{electrons} = \frac{Δ Q_{electrons}}{Δ t} = \frac{–0 . 300 \times 10^{- 3} C}{s} .

(9.8.5)

We divide this by the charge per electron, so that

\begin{array}{l} \frac{e^{–}}{s} & = & \frac{–0 . 300 \times 10^{- 3} C}{s} \times \frac{1 e^{–}}{–1 .60 \times 10^{- 19} C} \\ = & 1.88 \times 10^{15} \frac{e^{–}}{s} . \end{array}

(9.8.6)

Discussion

There are so many charged particles moving, even in small currents, that individual charges are not noticed, just as individual water molecules are not noticed in water flow. Even more amazing is that they do not always keep moving forward like soldiers in a parade. Rather they are like a crowd of people with movement in different directions but a general trend to move forward. There are lots of collisions with atoms in the metal wire and, of course, with other electrons.

Section Summary

Electric current $I$ is the rate at which charge flows, given by
$I = \frac{Δ Q}{Δ t},$
(9.8.7)
where $Δ Q$ is the amount of charge passing through an area in time $Δ t$ .
The direction of conventional current is taken as the direction in which positive charge moves.
The SI unit for current is the ampere (A), where $1 A = 1 C/s.$
Current is the flow of free charges, such as electrons and ions.

Conceptual Questions

Exercise 30

Can a wire carry a current and still be neutral—that is, have a total charge of zero? Explain.

Exercise 31

Car batteries are rated in ampere-hours (

A \cdot h

). To what physical quantity do ampere-hours correspond (voltage, charge, . . .), and what relationship do ampere-hours have to energy content?

Exercise 32

Why are two conducting paths from a voltage source to an electrical device needed to operate the device?

Exercise 33

In cars, one battery terminal is connected to the metal body. How does this allow a single wire to supply current to electrical devices rather than two wires?

Exercise 34

Why isn’t a bird sitting on a high-voltage power line electrocuted? Contrast this with the situation in which a large bird hits two wires simultaneously with its wings.

Problems & Exercises

Exercise 101

What is the current in milliamperes produced by the solar cells of a pocket calculator through which 4.00 C of charge passes in 4.00 h?

Solution

0.278 mA

Exercise 102

A total of 600 C of charge passes through a flashlight in 0.500 h. What is the average current?

Exercise 103

What is the current when a typical static charge of

0.250 μ C

moves from your finger to a metal doorknob in

1.00 μ s

Solution

0.250 A

Exercise 104

Find the current when 2.00 nC jumps between your comb and hair over a

0 . 500 - μ s

time interval.

Exercise 105

A large lightning bolt had a 20,000-A current and moved 30.0 C of charge. What was its duration?

Solution

1.50ms

Exercise 106

The 200-A current through a spark plug moves 0.300 mC of charge. How long does the spark last?

Exercise 107

(a) A defibrillator passes 12.0 A of current through the torso of a person for 0.0100 s. How much charge moves? (b) How many electrons pass through the wires connected to the patient? (See figure two problems earlier.)

Solution

(a) 0.120 C

(b)

7.50 \times 10^{17} electrons

Exercise 108

A clock battery wears out after moving 10,000 C of charge through the clock at a rate of 0.500 mA. (a) How long did the clock run? (b) How many electrons per second flowed?

Exercise 109

The batteries of a submerged non-nuclear submarine supply 1000 A at full speed ahead. How long does it take to move Avogadro’s number (

6.02 \times 10^{23}

) of electrons at this rate?

Solution

96.3 s

Exercise 110

Electron guns are used in X-ray tubes. The electrons are accelerated through a relatively large voltage and directed onto a metal target, producing X-rays. (a) How many electrons per second strike the target if the current is 0.500 mA? (b) What charge strikes the target in 0.750 s?

Exercise 111

A large cyclotron directs a beam of

{He}^{++}

nuclei onto a target with a beam current of 0.250 mA. (a) How many

{He}^{++}

nuclei per second is this? (b) How long does it take for 1.00 C to strike the target? (c) How long before 1.00 mol of

{He}^{++}

nuclei strike the target?

Solution

(a)

{7.81 × 10}^{14} {He}^{++} nuclei/s

(b)

{4.00 × 10}^{3} s

(c)

{7.71 × 10}^{8} s

electric current

the rate at which charge flows, I = ΔQ/Δt

ampere

(amp) the SI unit for current; 1 A = 1 C/s