Electric Current, Resistance, and Ohm’s Law
19 Current
Learning Objectives
- Define electric current, the ampere, and drift velocity.
- Explain the direction of conventional current.
- Use drift velocity to calculate electric current and vice versa.
Electric Current
Electric current is the rate at which electric charge flows through a material. A large current—such as the one used to start a truck engine—moves a large amount of charge in a short time. A much smaller current is sufficient to operate devices such as handheld calculators, hearing aids, or medical sensors.
Current is measured by determining how much charge passes through a cross-sectional area of a conductor during a given time interval. In equation form, electric current I is defined as
[latex]\mathbf{I}=\frac{\Delta Q}{\Delta t}[/latex]
where [latex]\Delta Q[/latex] is the amount of charge that passes through the conductor and [latex]\Delta t[/latex] is the corresponding time interval. (Often the initial time is taken as zero, so [latex]\Delta t=t[/latex].) This situation is illustrated in Figure 19.1.
The SI unit of current is the ampere (A), named after the French physicist André-Marie Ampère (1775–1836). Since current is charge divided by time, one ampere is defined as one coulomb of charge flowing each second:
[latex]1\ \text{A}=1\ \text{C/s}[/latex]
Current ratings appear throughout everyday life and healthcare. Household circuit breakers are rated by the maximum current they can safely carry, batteries and chargers specify the current they can supply, and medical devices such as infusion pumps, patient monitors, MRI systems, and defibrillators all operate using carefully controlled electric currents.
Example 19.1: Current in a Truck Battery and a Handheld Calculator
- What current is produced when a truck battery moves 720 C of charge in 4.00 s while starting the engine?
- How long does it take 1.00 C of charge to flow through a handheld calculator if the current is 0.300 mA?
Strategy
Both parts use the definition of electric current:
In part (a), the charge and time are known, so we calculate the current directly. In part (b), the charge and current are given, so we rearrange the equation to solve for the time required for that amount of charge to flow.
Solution (a)
Substitute the given values into the definition of current:
Discussion (a)
A current of 180 A is very large, but it is typical for the short time required to start an automobile engine. Starter motors must provide enough torque to overcome friction and begin turning the engine, which requires a substantial flow of electric charge.
Solution (b)
Rearrange the current equation to solve for time:
Discussion (b)
This corresponds to about 56 minutes. Although the calculator operates with a very small current, it still functions immediately because it requires very little electrical power. In contrast, starting a truck engine requires a much larger current to deliver energy quickly.
Figure 19.2 shows a simple electric circuit consisting of a battery, conducting wires, and a load (represented here by a resistor). A complete conducting path allows electric charge to move continuously around the circuit. Although real circuits can be much more complex, they are analyzed using the same basic principles introduced here.
Engineers and physicists usually represent circuits using schematic diagrams rather than realistic drawings. Schematics emphasize the electrical connections instead of the physical appearance of the components. The same schematic shown in Figure 19.2(b) could represent many different devices—from a flashlight or truck headlight to a medical instrument—because the physics governing current and energy transfer is the same in every case.
Notice that the current direction shown in Figure 19.2 is from the battery's positive terminal toward its negative terminal. This direction is called conventional current and is defined as the direction that positive charge would move through the circuit.
The actual moving charges depend on the material. In metal wires, current is carried almost entirely by electrons, which have negative charge and therefore move in the direction opposite to conventional current. In biological fluids, however, electric current is carried primarily by positive and negative ions moving simultaneously. This ionic current is responsible for electrical signaling in nerves, muscles, and the heart.
A conductor carrying current also contains an electric field (see Figure 19.3). This is an important difference from electrostatics. In electrostatic equilibrium, the electric field inside a conductor is zero. When current flows, however, the conductor is no longer in electrostatic equilibrium. The electric field inside the conductor provides the force that keeps the charge carriers moving and transfers energy throughout the circuit.
Take-Home Experiment: Modeling Electric Current
Find a drinking straw and a handful of small peas, beads, or other objects that can move freely through it. Lay the straw flat on a table and fill it completely. Then gently push one pea into one end of the straw. Notice that almost immediately a different pea is pushed out of the opposite end.
This simple demonstration provides a useful analogy for electric current. Although each individual pea moves only a short distance, the effect of your push is transmitted through the entire line almost instantly, causing another pea to emerge from the far end. Similarly, in a metal wire the electrons already occupy the conductor. When an electric field is established, they all begin drifting together, producing current throughout the circuit without requiring electrons to travel all the way from the battery to the device.
Think about the following questions:
- What represents the charge carriers (electrons or ions)?
- What represents the battery or energy source?
- How does this analogy help explain why a light turns on almost immediately after a switch is closed?
- Can you think of another everyday analogy for electric current?
Limitations of the analogy: The peas move because they physically collide with one another. Electric charges, however, move because electric forces produced by an electric field act throughout the conductor. Like all analogies, this one captures some aspects of the physics while simplifying others.
Example 19.2: How Many Electrons Flow Through a Calculator Each Second?
A handheld calculator operates with a current of [latex]0.300\ \text{mA}[/latex]. Assuming the current is carried by electrons, how many electrons pass through the circuit each second?
Strategy
Current is the rate at which charge flows:
[latex]\mathbf{I}=\frac{\Delta Q}{\Delta t}[/latex]
Each electron carries a charge of
[latex]e=1.60\times10^{-19}\ \text{C}[/latex]
To determine the number of electrons passing a point each second, divide the total charge flowing each second by the charge carried by one electron. Since we are counting particles, we use the magnitude of the electron charge.
Solution
Convert the current to amperes:
[latex]\mathbf{I}=0.300\ \text{mA}=0.300\times10^{-3}\ \text{A}[/latex]
Now calculate the number of electrons flowing each second:
[latex]\frac{\text{electrons}}{\text{s}}=\frac{0.300\times10^{-3}\ \text{C/s}}{1.60\times10^{-19}\ \text{C/electron}}[/latex]
[latex]=1.88\times10^{15}\ \text{electrons/s}[/latex]
Discussion
Although a calculator draws only a very small current, nearly two quadrillion electrons pass through the circuit every second. This enormous number results from the extremely small charge carried by each individual electron.
It is important to remember that these electrons do not race through the wire at high speed. Each electron moves only very slowly on average while undergoing countless collisions with atoms in the conductor. The electrical signal, however, propagates through the circuit almost immediately because the electric field is established throughout the conductor. In the following passage, we will quantify this slow average motion using the concept of drift velocity.
Drift Velocity
When you flip a light switch, the bulb turns on almost instantly, yet the electrons inside the wire do not travel from the switch to the bulb anywhere near the speed of light. This apparent contradiction is resolved by distinguishing between the speed of the electrical signal and the speed of the individual charge carriers. The electrical signal—a change in the electric field throughout the circuit—propagates extremely rapidly, while the individual electrons move only very slowly on average.
In many circuits, the electrical signal travels at speeds on the order of [latex]10^{8}\ \text{m/s}[/latex], a significant fraction of the speed of light. Surprisingly, the average speed of the electrons themselves, called the drift velocity, is typically only about [latex]10^{-4}\ \text{m/s}[/latex].
The rapid signal speed occurs because electric forces act quickly throughout the conductor. When electrons are pushed into one end of a wire, they repel neighboring electrons, which in turn repel electrons farther down the wire. Because charge cannot easily accumulate inside a conductor, electrons begin leaving the far end almost immediately. More precisely, what travels rapidly is the change in the electric field established throughout the conductor.
Good conductors contain large numbers of mobile charge carriers. In metals, these carriers are free electrons. As shown in Figure 19.5, individual electrons undergo frequent collisions with atoms in the metal, so their motion is largely random—similar to the random motion of molecules in a gas. When an electric field is applied, however, this random motion acquires a small net motion in one direction. The average velocity of the charge carriers is called the drift velocity, [latex]v_d[/latex]. For electrons, the drift direction is opposite the electric field and opposite the direction of conventional current.
Conduction of Electricity and Heat
Materials that are good electrical conductors are usually good thermal conductors because their mobile electrons transport both electric charge and thermal energy. Metals such as copper, silver, and aluminum are excellent conductors of both electricity and heat.
Collisions between free electrons and the atoms of the conductor transfer energy to the material. Although the electric field continually accelerates the electrons, repeated collisions prevent them from continuously speeding up. Instead, the energy supplied by the electric field is transferred to the conductor as thermal energy, often raising its temperature. This is why a continuous power source is required to maintain a steady current. An important exception is a superconductor, which can carry current with essentially no electrical energy loss.
Take-Home Experiment: Examining a Light Bulb Filament
Find an incandescent light bulb with a visible filament. Examine the filament carefully and describe its shape. To what parts of the bulb is it connected? Why do you think the filament is wound into a tight coil instead of being a straight wire?
We can now combine the definition of current with a simple counting argument to derive a useful relationship between current and drift velocity. Consider the segment of wire shown in Figure 19.6. Let [latex]n[/latex] represent the number of free charge carriers per unit volume (the carrier density). A segment of wire with cross-sectional area [latex]A[/latex] and length [latex]x[/latex] has a volume of [latex]Ax[/latex], so it contains [latex]nAx[/latex] charge carriers. If each carrier has charge [latex]q[/latex], then the total charge contained in the segment is
[latex]\Delta Q=nqAx[/latex]
If all of this charge moves through the wire in a time interval [latex]\Delta t[/latex], then
[latex]\mathbf{I}=\frac{\Delta Q}{\Delta t}=\frac{nqAx}{\Delta t}[/latex]
Since the carriers move an average distance [latex]x[/latex] during the time interval [latex]\Delta t[/latex], the magnitude of the drift velocity is
[latex]v_d=\frac{x}{\Delta t}[/latex]
Substituting this expression into the current equation gives the important relationship
[latex]\mathbf{I}=nqAv_d[/latex]
Current and Drift Velocity
[latex]\mathbf{I}=nqAv_d[/latex]
where
- I = current
- n = charge carrier density (number of carriers per unit volume)
- q = charge of each carrier
- A = cross-sectional area of the conductor
- vd = drift velocity of the charge carriers
Drift velocity is only part of the story. Individual electrons move much faster than [latex]v_d[/latex] because of their random thermal motion. The drift velocity is simply the small average motion produced by the electric field. In metals, only the loosely bound free electrons participate in electrical conduction. In insulators, electrons remain tightly bound to their atoms and cannot move freely, which is why these materials conduct electricity very poorly.
This distinction explains why metals such as copper and aluminum are excellent electrical conductors, whereas materials such as rubber, glass, and most plastics are effective electrical insulators.
Example 19.3: Calculating the Drift Velocity in a Copper Wire
A 12-gauge copper wire (diameter 2.053 mm) carries a current of 20.0 A. Assuming each copper atom contributes one free electron to electrical conduction, calculate the average drift velocity of the electrons. The density of copper is
Strategy
We use the relationship
where:
- I is the current,
- n is the number of free electrons per unit volume,
- q is the magnitude of the electron charge,
- A is the cross-sectional area of the wire, and
- vd is the drift velocity.
The current is known, so we first calculate the electron density [latex]n[/latex] from the properties of copper, then determine the wire's cross-sectional area, and finally solve for the drift velocity.
Solution
Step 1: Determine the free-electron density.
Assuming one free electron per copper atom,
[latex]\begin{aligned} n &=\left(\frac{1\ \text{electron}}{\text{atom}}\right) \left(\frac{6.02\times10^{23}\ \text{atoms}}{\text{mol}}\right) \left(\frac{1\ \text{mol}}{63.54\ \text{g}}\right) \left(\frac{1000\ \text{g}}{1\ \text{kg}}\right) \left(\frac{8.80\times10^{3}\ \text{kg}}{1\ \text{m}^3}\right)\\ &=8.34\times10^{28}\ \text{electrons/m}^3. \end{aligned}[/latex]
Step 2: Calculate the wire's cross-sectional area.
The radius is
[latex]r=\frac{2.053\times10^{-3}\ \text{m}}{2}=1.0265\times10^{-3}\ \text{m}[/latex]
Therefore,
[latex]\begin{aligned} A&=\pi r^2\\ &=\pi(1.0265\times10^{-3}\ \text{m})^2\\ &=3.31\times10^{-6}\ \text{m}^2. \end{aligned}[/latex]
Step 3: Solve for the drift velocity.
Using the magnitude of the electron charge,
[latex]q=1.60\times10^{-19}\ \text{C}[/latex]
we obtain
[latex]\begin{aligned} v_d &=\frac{\mathbf{I}}{nqA}\\ &=\frac{20.0\ \text{A}} {(8.34\times10^{28}\ \text{m}^{-3}) (1.60\times10^{-19}\ \text{C}) (3.31\times10^{-6}\ \text{m}^2)}\\ &=4.53\times10^{-4}\ \text{m/s}. \end{aligned}[/latex]
Discussion
The electrons drift at an average speed of only
[latex]4.53\times10^{-4}\ \text{m/s}[/latex]
which is less than half a millimeter per second. Although this motion is extremely slow, the electrical signal travels through the circuit at nearly the speed of light because the electric field is established throughout the conductor almost immediately.
Remember that electrons move opposite the direction of the electric field and opposite the direction of conventional current. The value calculated here is the magnitude of the drift velocity.
Section Summary
- Electric current I is the rate at which electric charge flows:
[latex]\mathbf{I}=\frac{\Delta Q}{\Delta t}[/latex]
where [latex]\Delta Q[/latex] is the amount of charge passing through a cross-sectional area during the time interval [latex]\Delta t[/latex].
- The direction of conventional current is defined as the direction that positive charge would move, even though electrons move in the opposite direction in metallic conductors.
- The SI unit of current is the ampere (A):
[latex]1\ \text{A}=1\ \text{C/s}[/latex]
- Electric current can be carried by different types of charge carriers, including free electrons in metals and positive and negative ions in biological fluids.
- The drift velocity, [latex]v_d[/latex], is the average velocity of the charge carriers in a conductor.
- Current, drift velocity, and the properties of the conductor are related by
[latex]\mathbf{I}=nqAv_d[/latex]
where [latex]n[/latex] is the charge carrier density, [latex]q[/latex] is the magnitude of the charge carried by each particle, and [latex]A[/latex] is the conductor's cross-sectional area.
- Although individual charge carriers drift very slowly, changes in the electric field—and therefore electrical signals—propagate through circuits extremely rapidly.
Conceptual Questions
- Can a wire carry a current and still be neutral—that is, have a total charge of zero? Explain.
- Car batteries are rated in ampere-hours ([latex]\text{A}\cdot \text{h}[/latex]). To what physical quantity do ampere-hours correspond (voltage, charge, . . .), and what relationship do ampere-hours have to energy content?
- If two different wires having identical cross-sectional areas carry the same current, will the drift velocity be higher or lower in the better conductor? Explain in terms of the equation [latex]{v}_{\text{d}}=\frac{\mathbf{I}}{\text{nqA}}[/latex] by considering how the density of charge carriers [latex]n[/latex] relates to whether or not a material is a good conductor.
- Why are two conducting paths from a voltage source to an electrical device needed to operate the device?
- 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?
- 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
- 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?
- A total of 600 C of charge passes through a flashlight in 0.500 h. What is the average current?
- What is the current when a typical static charge of [latex]0\text{.}\text{250}\phantom{\rule{0.25em}{0ex}}\mu \text{C}[/latex] moves from your finger to a metal doorknob in [latex]1.00\phantom{\rule{0.25em}{0ex}}\mu \text{s}[/latex]?
- Find the current when 2.00 nC jumps between your comb and hair over a [latex]0\text{.}\text{500 -}\phantom{\rule{0.25em}{0ex}}\mu \text{s}[/latex] time interval.
- A large lightning bolt had a 20,000-A current and moved 30.0 C of charge. What was its duration?
- The 200-A current through a spark plug moves 0.300 mC of charge. How long does the spark last?
- A defibrillator sends a 6.00-A current through the chest of a patient by applying a 10,000-V potential as in the figure below.
- What is the resistance of the path?
- The defibrillator paddles make contact with the patient through a conducting gel that greatly reduces the path resistance. Discuss the difficulties that would ensue if a larger voltage were used to produce the same current through the patient, but with the path having perhaps 50 times the resistance. (Hint: The current must be about the same, so a higher voltage would imply greater power. Use this equation for power: [latex]P={\mathbf{I}}^{2}R[/latex].)
Figure 19.7: The capacitor in a defibrillation unit drives a current through the heart of a patient. - During open-heart surgery, a defibrillator can be used to bring a patient out of cardiac arrest. The resistance of the path is [latex]5\text{00 Ω}[/latex] and a 10.0-mA current is needed. What voltage should be applied?
- A defibrillator passes 12.0 A of current through the torso of a person for 0.0100 s.
- How much charge moves?
- How many electrons pass through the wires connected to the patient? (See Figure 19.7.)
- A clock battery wears out after moving 10,000 C of charge through the clock at a rate of 0.500 mA.
- How long did the clock run?
- How many electrons per second flowed?
- 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 ([latex]6\text{.}\text{02}×{\text{10}}^{\text{23}}[/latex]) of electrons at this rate?
- 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.
- How many electrons per second strike the target if the current is 0.500 mA?
- What charge strikes the target in 0.750 s?
- A large cyclotron directs a beam of [latex]{\text{He}}^{\text{++}}[/latex] nuclei onto a target with a beam current of 0.250 mA.
- How many [latex]{\text{He}}^{\text{++}}[/latex] nuclei per second is this?
- How long does it take for 1.00 C to strike the target?
- How long before 1.00 mol of [latex]{\text{He}}^{\text{++}}[/latex] nuclei strike the target?
- Repeat the above Example 19.3, but for a wire made of silver and given there is one free electron per silver atom.
- Using the results of the above Example 19.3, find the drift velocity in a copper wire of twice the diameter and carrying 20.0 A.
- A 14-gauge copper wire has a diameter of 1.628 mm. What magnitude current flows when the drift velocity is 1.00 mm/s? (See Example 19.3 for useful information.)
- SPEAR, a storage ring about 72.0 m in diameter at the Stanford Linear Accelerator (closed in 2009), has a 20.0-A circulating beam of electrons that are moving at nearly the speed of light (see Figure 19.8.) How many electrons are in the beam?
Glossary
- electric current
- The rate at which electric charge flows through a conductor, defined by [latex]I=\Delta Q/\Delta t[/latex].
- ampere (A)
- The SI unit of electric current. One ampere is equal to one coulomb of charge passing a point each second ([latex]1\ \text{A}=1\ \text{C/s}[/latex]).
- drift velocity
- The average velocity of charge carriers as they move through a material in response to an electric field.
- conventional current
- The direction in which positive charge would move. In metallic conductors, this direction is opposite the motion of electrons.
The rate at which electric charge flows through a conductor, defined by [latex]I=\Delta Q/\Delta t[/latex].
The SI unit of electric current. One ampere is equal to one coulomb of charge passing a point each second ([latex]1\ \text{A}=1\ \text{C/s}[/latex]).
The average velocity of charge carriers as they move through a material in response to an electric field.
The direction in which positive charge would move. In metallic conductors, this direction is opposite the motion of electrons.
