In 1799 Alessandro Volta is credited with building the first electric battery.
Volta's battery produced voltage with an electrolyte sandwiched between two different metals.
In 1831 Michael Faraday discovered that voltage is produced by moving a magnet past a wire. Coal, gas,
nuclear, hydroelectric, and geothermal power plants all produce electricity by moving magnets past wires.
Solar panels are growing in popularity as an alternative voltage source.
They use the photovoltaic effect to convert light into
electricity.
Albert Einstein explained the photovoltaic effect in 1905, but it wasn't until the 1950s that Bell
Laboratories developed the first solar panels.
Electric Current
Voltage sources produce electric current in conductors.
Electric current is the average motion of a huge number of charges through a conductor.
The electrons don't just move in a straight line.
They drift through the conductor in a random walk bouncing off other charges.
These simulations model electric current with Coulomb's law. They don't include the nuances of quantum mechanics.
A small imbalance in net charge produces an electric field. The electric field points in the same direction
as the current.
In 1746 Benjamin Franklin mistakenly assigned a negative value to the
charge carriers that we now call electrons.
This annoying choice set a precedent that made the direction of electric current opposite of the direction
the electrons actually move.
Negative charge flows away from negative potential and towards positive.
Positive charge does the opposite.
Question: What direction is the current in the above simulations?
answer
Current is flowing to the left.
Current is defined as the flow of positive charge. We see the electrons move to the
right, but they have negative charge. The current is in the opposite direction of the electron flow.
Question: Why do the electrons move, but not the nuclei?
answer
$$\text{electron} = 9.1 \times 10^{-31} \, \mathrm{kg}$$
$$\text{copper nucleus} = 1.0 \times 10^{-25} \, \mathrm{kg}$$
An electron is about 100 000 times lighter than a typical nucleus. This means forces
experienced by electrons will produce larger accelerations and larger velocities.
Additionally, the nucleus and the lower electron shells are strongly held in place by the metallic
bonds. Metallic bonds are formed when the outer electron shells are shared between nuclei. This
outer electron shell forms a conductive layer in metals.
The average flow of electrons in a typical wire is around 0.0002 m/s.
Question: If the flow of electrons is slow, why isn't there a delay in the
music coming out of my headphones? Why is communication by phone, TV, and internet near instantaneous?
answer
In conductors, an imbalance in electric charge for one atom will cause nearby atoms to have the same
imbalance.
Changes in this imbalance of charge spread as a wave at near the speed of light.
Electric signals are waves in the electric field.
The electrons aren't the signal. The signal is the changes in the electric and magnetic
field.
Current measures the flow of charge over a period of time. Current is not a substance.
Current is a rate, like dollars per hour is a rate.
$$I = \frac{q}{\Delta t}$$
\(I\) = electric current [A, amps, amperes, C/s]
\(q\) = charge that passes a point on a wire [C, coulombs]
\(\Delta t\) = a period of time [s]
It can be helpful, and accurate, to think of electric current like water current. Free electrons flow through
metals like H₂O flows through pipes.
Example: 1.2 coulombs of charge passes through a typical LED every minute. What is the current in
the LED?
solution
$$I = \frac{q}{\Delta t}$$
$$I = \frac{1.2\, \mathrm{C}}{60\, \mathrm{s}}$$
$$ I= 0.02 \, \mathrm{\tfrac{C}{s}}$$
$$ I= 0.02 \, \mathrm{A}$$
Example: Two wires of cross-sectional area 1.6 mm² connect the terminals of a battery to
the circuitry in a clock. After 0.04 seconds, 5.0 × 10¹⁴ electrons move through a point on the wire. What is
the current in the wires?
strategy
Build a conversion fraction to convert 1 electron into electric charge. Use the charge and the time
to calculate the current.
Example: If 0.320 mA of charge flow through a calculator, how many electrons pass through
per second?
solution
$$ I = \frac{q}{\Delta t}$$
$$ q = I\Delta t$$
$$ q = (0.320 \times 10^{-3}\, \mathrm{A}) (1\, \mathrm{s})$$
$$ q = 0.000320 \, \mathrm{C} \left(\frac{1 \, e^{-}} {1.6 \times 10^{-19}\, \mathrm{C}}\right) $$
$$ q = 2 \times 10^{15} \, e^{-}$$
Voltage (Potential Difference)
Voltage can be thought of as a measure of electric charge density.
Technically, voltage measures the difference in electrical potential between two
points.
Voltage is defined so that negative charges are pulled towards positive potential, and positive charges are
pulled towards negative potential.
The two terminals in a wall socket are sources of voltage.
The positive and negative terminals on a battery also provide voltage.
These simulations model a "circuit" with a "battery". The battery pushes the negative charges down,
which produces a difference in electric potential.
Areas with more negative charge have negative potential.
Areas with more positive charge have positive potential.
Question: Why does only the right simulation produce a flow of charges.
answer
When the circuit isn't connected electric charge builds up at each end point. Connecting the
ends to form a loop guides the charges back to the start to complete another cycle.
Question: Where is there a higher concentration of electrons?
answer
The bottom, where the color is more blue. This happens in both simulations because the battery
is pushing electrons down.
Measure: Let each sim run until they settle on a voltage. Record that voltage.
results
The left simulation has a voltage around 7.2 V.
The right simulation has a voltage around 4.6 V.
Follow Up Question: Why is the voltage lower for the circuit on the right?
answer
Connecting the ends of the circuit gives the charges a conductive path. This path allows
some of the built up potential to equalize.
Electric circuits have either direct current or alternating current depending on the source of the voltage.
In DC (direct current)circuits,
a constant voltage source pushes the electrons around the circuit in one direction.
Batteries are a common source of DC.
In AC (alternating current)circuits,
a rapidly changing potential pushes the electrons forward and backwards about 60 times a second.
A wall socket is a source of AC.
Question: Predict what happens to the flow of electrons after adding resistors.
answer
Adding more resistors will cause the electron's average forward progress to be slower because the
resistors cause "traffic jams".
Question: What would happen if we removed all the resistors?
answer
Removing most of the resistance in a circuit leads to a short circuit.
This causes the average electron flow to increase, producing dangerous amounts of heat.
A DC voltage source is represented by one or more pairs of lines.
The longer line in the symbol represents the positive terminal, and it is called the
cathode. The shorter line is the negative terminal, called the anode.
Electrons flow out of the anode, but the current is defined as coming out of the cathode.
Question: What's an example of a DC voltage source?
answer
Batteries, solar panels, and DC motors are DC voltage sources.
Wall sockets are sources of AC voltage.
Ohm's Law
In 1827, German physicist Georg Ohm published his work on the relationship between electric current and
voltage. Ohm discovered his law after measuring the voltage on wires of different length. He found that as
wires got longer they had less current for the same voltage.
Ohm's Law says that the voltage in a conductor is proportional to the current. This means we can build an
equation where V = I times a constant. The constant is called resistance and it measures
how current responds to a change in voltage.
The resistance of a material is the inverse of its conductivity. Conductors have nearly zero resistance and
insulators have very high resistance.
$$V = IR$$
\(V\) = voltage, a change in electric potential [V, volts]
Technically ΔV, but V is often used for simplicity.
\(I\) = electric current [A, amps, amperes]
\(R\) = resistance [Ω, ohms]
This form of Ohm's law is for direct current circuits only.
Circuit elements convert electric potential into other types of energy. LEDs make light. Electric motors
make motion. Electric speakers make sound. Logic circuits perform calculations.
Any element of a circuit that does something will cause the electric potential to drop and add resistance to
the circuit.
Play with this circuit simulator for resistance.
Adjust the wiggle of the resistor to see how it changes the voltage and current.
To run this circuit simulation click on the yellow V then slide the EMF setting to the right.
Semiconductive elements, called resistors,
have more resistance than a metal wire, but still much less resistance than an insulator, like air or plastic.
Resistors are added to a circuit for precise control over current. If the current is too high, the
circuit will get hot. If the current is too low the circuit can't do its job.
The IEC symbol for a resistor is this rectangle.
The American representation of a resistor is this squiggle, which I use because I like the way it looks.
A pipe with flowing water is analogous to a wire
with flowing charge.
A wire is like a water pipe.
An electric charge is like a water molecule.
A battery is like a water pump.
Electric current is like water flow.
Electric potential is like water pressure.
Question: What would a resistor be in this analogy?
answer
A resistor slows down the flow (current), and drops the pressure (potential). In a water pipe a
resistor would be a narrowing of the pipe, or a partial clog.
Example: Find the voltage for a resistor that has 300 Ω of resistance and a current of
0.05 A.
solution
$$ V = IR$$
$$ V = (0.05\, \mathrm{A}) (300 \, \Omega)$$
$$ V = 15 \, \mathrm{V}$$
Example: Find the current for a 100 Ω resistor with 2 V drop.
solution
$$ V = IR$$
$$ I = \frac{V}{R}$$
$$ I = \frac{2 \, \mathrm{V}}{100 \, \Omega}$$
$$ I = 0.02\, \mathrm{A}$$
Example: If I double the voltage in a circuit while keeping the resistance the same, what
happens to the current?
solution
Resistance is constant so we can just pretend it is one.
$$V=IR$$
$$V=I$$
$$2V=2I$$
Voltage and current are directly proportional.
Doubling V will double I.
resistance =
Ω
Question: What is the relationship between resistance, current, and voltage?
For constant resistance:
When voltage is increased, the current is __________.
When voltage is decreased, the current is __________.
answer
When voltage is increased, the current is increased.
When voltage is decreased, the current is decreased.
Voltage and current are directly proportional.
They increase and decrease together.
For constant voltage:
When resistance is increased, the current is __________.
When resistance is decreased, the current is __________.
answer
When resistance is increased, the current is decreased.
When resistance is decreased, the current is increased.
Resistance and current are inversely proportional.
When one increases the other decreases.
Example: What value resistor would drop 1.5 V in a 0.001 A current?
solution
$$ V = IR$$
$$ R = \frac{V}{I}$$
$$ R = \frac{1.5\, \mathrm{V}}{0.001\, \mathrm{A}}$$
$$ R = 1500 \, \Omega$$
Example: What voltage would cause 0.02 C of charge to pass through a 10 Ω
resistor every 10 s?
solution
$$ I = \frac{q}{\Delta t}$$
$$ I = \frac{0.02}{10}$$
$$ I = 0.002 \, \mathrm{A}$$
$$ V = IR $$
$$ V = (0.002)(10)$$
$$ V = 0.02 \, \mathrm{V}$$
In case you wanted more practice I used AI to make some more problems. The rest of the site I made by hand, but generating endless problems seemed safe. I did find mistakes in the AI generated problems, and there are probably some I didn't find. Let me know if something could be fixed. I also added a practice problem on each page with no solution. That's intentional. Have fun!
Unless a problem says otherwise, assume the wires and batteries are ideal and use conventional current for direction questions.
Question: Electrons drift to the right through a wire. Which way is the conventional current?
answer
Conventional current points to the left. Electron motion and conventional current are in opposite directions because electrons are negatively charged.
Example: A small sensor wire in a greenhouse monitor is sending data from a humidity probe. The probe checks the air every 10 seconds, but that timing is not needed here. If 3.6 C of charge passes through the wire in 2.0 minutes, what is the current?
solution
Convert minutes to seconds.
$$\Delta t = 120 \, \mathrm{s}$$
$$I = \frac{q}{\Delta t}$$
$$I = \frac{3.6 \, \mathrm{C}}{120 \, \mathrm{s}}$$
$$I = 0.030 \, \mathrm{A}$$
Example: During a brief pulse in a doorbell circuit, 0.480 C passes one point in 25 ms. The button cap is about 2.0 cm wide, but the current depends on charge and time. What current flows during the pulse?
solution
Convert milliseconds to seconds.
$$\Delta t = 0.025 \, \mathrm{s}$$
$$I = \frac{q}{\Delta t}$$
$$I = \frac{0.480 \, \mathrm{C}}{0.025 \, \mathrm{s}}$$
$$I = 19.2 \, \mathrm{A}$$
Question: A light turns on almost instantly when a switch is closed, but electrons drift through the wire slowly. How can both statements be true?
answer
The electric influence through the circuit spreads quickly when the switch is closed. The individual electrons still drift slowly; they do not need to travel from the switch all the way to the bulb before the bulb responds.
Example: A 1.0 m phone cable carries 250 mA for 30 s while a phone first starts charging. How much charge passes one point in the cable?
solution
Convert milliamps to amps.
$$I = 0.250 \, \mathrm{A}$$
$$I = \frac{q}{\Delta t}$$
$$q = I\Delta t$$
$$q = (0.250 \, \mathrm{A})(30 \, \mathrm{s})$$
$$q = 7.5 \, \mathrm{C}$$
Example: A small motor spins a plastic fan with 7 blades. During one test, 18 C of charge passes through the motor while the current is 0.75 A. How long does the motor run?
solution
$$I = \frac{q}{\Delta t}$$
$$\Delta t = \frac{q}{I}$$
$$\Delta t = \frac{18 \, \mathrm{C}}{0.75 \, \mathrm{A}}$$
$$\Delta t = 24 \, \mathrm{s}$$
Example: About 2.5 × 1015 electrons pass one point in a wire in 0.20 s. What is the magnitude of the conventional current?
solution
Use the charge of one electron as a magnitude.
$$q = ne$$
$$q = (2.5 \times 10^{15})(1.60 \times 10^{-19} \, \mathrm{C})$$
$$q = 4.0 \times 10^{-4} \, \mathrm{C}$$
$$I = \frac{q}{\Delta t}$$
$$I = \frac{4.0 \times 10^{-4} \, \mathrm{C}}{0.20 \, \mathrm{s}}$$
$$I = 2.0 \times 10^{-3} \, \mathrm{A}$$
This is 2.0 mA.
Example: A calculator draws 0.80 mA while showing an 8-digit answer on its display. About how many electrons pass one point each second?
solution
Convert milliamps to amps.
$$I = 0.00080 \, \mathrm{A}$$
$$q = I\Delta t$$
$$q = (0.00080 \, \mathrm{A})(1.0 \, \mathrm{s})$$
$$q = 8.0 \times 10^{-4} \, \mathrm{C}$$
$$n = \frac{q}{e}$$
$$n = \frac{8.0 \times 10^{-4} \, \mathrm{C}}{1.60 \times 10^{-19} \, \mathrm{C}}$$
$$n = 5.0 \times 10^{15}$$
About 5.0 × 1015 electrons pass each second.
Question: A battery is connected to a wire, but the wire is broken at one point. Can there still be voltage across the break? Can there be steady current around the circuit?
answer
There can be voltage across the break because the battery can still create an electric potential difference. There is no steady current around the circuit because the path is not complete.
Example: A 9.0 V battery is connected across a 330 Ω resistor on a breadboard. The jumper wires are different colors, but assume they are ideal. What current flows through the resistor?
solution
$$V = IR$$
$$I = \frac{V}{R}$$
$$I = \frac{9.0 \, \mathrm{V}}{330 \, \Omega}$$
$$I = 0.027 \, \mathrm{A}$$
This is 27 mA.
Example: A small lamp uses 20 mA from a 1.5 V battery. What is the lamp's resistance?
solution
Convert milliamps to amps.
$$I = 0.020 \, \mathrm{A}$$
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{1.5 \, \mathrm{V}}{0.020 \, \mathrm{A}}$$
$$R = 75 \, \Omega$$
Example: A current of 18 mA flows through a 470 Ω resistor. What voltage is across the resistor?
solution
Convert milliamps to amps.
$$I = 0.018 \, \mathrm{A}$$
$$V = IR$$
$$V = (0.018 \, \mathrm{A})(470 \, \Omega)$$
$$V = 8.46 \, \mathrm{V}$$
Example: An indicator light needs 3.0 mA when it has 9.0 V across it. What resistance would give that current?
solution
Convert milliamps to amps.
$$I = 0.0030 \, \mathrm{A}$$
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{9.0 \, \mathrm{V}}{0.0030 \, \mathrm{A}}$$
$$R = 3000 \, \Omega$$
This is 3.0 kΩ.
Example: A 220 Ω resistor is connected to a battery for 5.0 s. What current flows through it?
answer
This cannot be solved from the information given. Ohm's law needs the voltage and the resistance. The time does not replace the missing voltage.
Example: A sensor should have 2.5 mA of current when 5.0 V is across it. What resistance should the sensor have?
solution
$$I = 0.0025 \, \mathrm{A}$$
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{5.0 \, \mathrm{V}}{0.0025 \, \mathrm{A}}$$
$$R = 2000 \, \Omega$$
This is 2.0 kΩ.
Example: A low-current display mounted behind a 4 cm window has 3.3 V across it and draws 0.40 mA. What is its resistance?
solution
Convert milliamps to amps.
$$I = 0.00040 \, \mathrm{A}$$
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{3.3 \, \mathrm{V}}{0.00040 \, \mathrm{A}}$$
$$R = 8250 \, \Omega$$
This is 8.25 kΩ.
Question: A resistor stays the same. What happens to the current if the voltage across it doubles? What happens if the voltage stays the same but the resistance doubles?
answer
If resistance stays the same, doubling voltage doubles current. If voltage stays the same, doubling resistance cuts the current in half.
Example: A portable radio uses a 12 V battery pack connected across a 680 Ω test resistor for 3.0 minutes. The battery pack holds eight small cells, but treat the voltage as 12 V. How much charge passes through the resistor?
solution
First find the current.
$$I = \frac{V}{R}$$
$$I = \frac{12 \, \mathrm{V}}{680 \, \Omega}$$
$$I = 0.0176 \, \mathrm{A}$$
Convert minutes to seconds.
$$\Delta t = 180 \, \mathrm{s}$$
$$q = I\Delta t$$
$$q = (0.0176 \, \mathrm{A})(180 \, \mathrm{s})$$
$$q = 3.2 \, \mathrm{C}$$
Example: A 5.0 V source is across a 1.5 kΩ resistor for 10 minutes. How much charge passes through the resistor?
solution
Convert kilo-ohms to ohms.
$$R = 1500 \, \Omega$$
$$I = \frac{V}{R}$$
$$I = \frac{5.0 \, \mathrm{V}}{1500 \, \Omega}$$
$$I = 0.0033 \, \mathrm{A}$$
Convert minutes to seconds.
$$\Delta t = 600 \, \mathrm{s}$$
$$q = I\Delta t$$
$$q = (0.0033 \, \mathrm{A})(600 \, \mathrm{s})$$
$$q = 2.0 \, \mathrm{C}$$
Example: A 2.2 kΩ resistor has a current of 1.2 mA. What voltage is across it, and how much charge passes in 45 s?
solution
Convert the units first.
$$R = 2200 \, \Omega$$
$$I = 0.0012 \, \mathrm{A}$$
Find the voltage.
$$V = IR$$
$$V = (0.0012 \, \mathrm{A})(2200 \, \Omega)$$
$$V = 2.64 \, \mathrm{V}$$
Now find the charge.
$$q = I\Delta t$$
$$q = (0.0012 \, \mathrm{A})(45 \, \mathrm{s})$$
$$q = 0.054 \, \mathrm{C}$$
Example: An unknown resistor has 6.0 V across it and 0.015 A through it. What is the resistance?
solution
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{6.0 \, \mathrm{V}}{0.015 \, \mathrm{A}}$$
$$R = 400 \, \Omega$$
Example: A toy motor draws 0.60 A from a 3.0 V battery while lifting a small paper flag. What is its effective resistance while running?
solution
$$V = IR$$
$$R = \frac{V}{I}$$
$$R = \frac{3.0 \, \mathrm{V}}{0.60 \, \mathrm{A}}$$
$$R = 5.0 \, \Omega$$
Example: A wire carries 75 mA for 2.0 minutes while a small display warms up and its brightness settles. The display has 12 tiny segments, but only the current and time matter here. How much charge passes one point in the wire?
solution
Convert milliamps to amps and minutes to seconds.
$$I = 0.075 \, \mathrm{A}$$
$$\Delta t = 120 \, \mathrm{s}$$
$$q = I\Delta t$$
$$q = (0.075 \, \mathrm{A})(120 \, \mathrm{s})$$
$$q = 9.0 \, \mathrm{C}$$
Example: A 24 V source is connected across a resistor. In 15 s, 1.8 C of charge passes through the resistor. What is the resistance?
solution
Find the current first.
$$I = \frac{q}{\Delta t}$$
$$I = \frac{1.8 \, \mathrm{C}}{15 \, \mathrm{s}}$$
$$I = 0.12 \, \mathrm{A}$$
Now use Ohm's Law.
$$R = \frac{V}{I}$$
$$R = \frac{24 \, \mathrm{V}}{0.12 \, \mathrm{A}}$$
$$R = 200 \, \Omega$$
Example: A resistor should allow 0.050 C to pass in 2.0 s when connected to a 6.0 V source. What resistance is needed?
solution
Find the current needed.
$$I = \frac{q}{\Delta t}$$
$$I = \frac{0.050 \, \mathrm{C}}{2.0 \, \mathrm{s}}$$
$$I = 0.025 \, \mathrm{A}$$
Use the voltage to find resistance.
$$R = \frac{V}{I}$$
$$R = \frac{6.0 \, \mathrm{V}}{0.025 \, \mathrm{A}}$$
$$R = 240 \, \Omega$$
Why must a simple circuit form a closed loop for a light bulb to stay lit?
answer
Charges need a complete path from one battery terminal, through the circuit, and back to the other terminal. An open gap interrupts that path, so current cannot continue.
In the article's water-flow comparison, what do voltage and resistance represent?
answer
Voltage is like the height difference that can drive water downhill. Resistance is like a narrow or blocked path that makes flow harder.
Why are power-cord wires covered in plastic instead of left as bare metal?
answer
The metal wire is a conductor, but plastic is an insulator. The plastic coating keeps current in the wire and helps protect people from electric shock.