This unit is about how voltage, current, and resistance work together in real circuits, from a single resistor to strings of bulbs and household wiring. By the end, you should be able to find the current, voltage, and power for any resistor in a series, parallel, or simple combination circuit.
Read the overview first, then work through each topic note and try its checkpoint question before you look at the answer.
Circuits connect ideas you already know about energy and charge to devices you use every day. This unit starts with Ohm's law, applies it to series and parallel wiring, and finishes with electric power and energy cost.
Honors Physics Unit 19 covers electrical circuits in four topics. You learn Ohm's law () and what current, voltage, and resistance mean. Then you analyze series circuits, where current is the same everywhere and voltages add, and parallel circuits, where voltage is the same across every branch and currents add. The unit ends with electric power in direct-current circuits. For a resistor, . Energy use is measured in joules and kilowatt-hours.
A battery's voltage is the energy it gives each coulomb of charge, measured in joules per coulomb. Resistance controls how much current that push produces. Ohm's law, , is the link you use in almost every problem.
Resistors in series share one current and split the voltage. Resistors in parallel share one voltage and split the current. Once you know which quantity is shared, the rest of the problem follows from Ohm's law.
A resistor turns electrical energy into heat and light at a rate of watts. Multiplying power by time gives energy, which is what the electric company bills you for in kilowatt-hours.
In a steady-state circuit, charge does not accumulate at a junction, so the current into it equals the current out. Around a closed loop, the signed voltage changes add to zero, consistent with conservation of energy. These conservation laws support the series, parallel, and power rules.
Ohm's law says the voltage across a resistor equals the current through it times its resistance, V = IR.
In a series circuit, components sit end to end on a single path, so the same current flows through every one of them.
In a parallel circuit, each component sits on its own branch between the same two junctions.
Electric power is the rate at which a circuit converts electrical energy into heat, light, or motion.
Ohm's law says the voltage across a resistor equals the current through it times its resistance, . Voltage is measured in volts (V), current in amperes (A), and resistance in ohms ().
Current is the rate charge flows past a point, . One ampere means one coulomb per second. Voltage, or potential difference, is the difference in electric potential energy per unit charge between two points. An ideal 9 V battery supplies 9 J of energy per coulomb of charge passing through it.
Resistance depends on the material, the length, and the thickness of the conductor. A longer, thinner wire has more resistance than a short, thick one made of the same metal. Copper has low resistance, while rubber and plastic have very high resistance.
Ohm's law describes ohmic materials, where resistance stays constant. For those, a graph of current versus voltage is a straight line through the origin. A light bulb filament is non-ohmic because its resistance rises as it heats up.
Worked example. A 9 V battery is connected to a 450 resistor. The current is A, or 20 mA. In 60 s, the charge that flows is C.
Check yourself: If you double the voltage across a fixed 10 resistor from 5 V to 10 V, what happens to the current? (Answer: It doubles, from 0.5 A to 1 A, because and R didn't change.)
In a series circuit, components sit end to end on a single path, so the same current flows through every one of them. The voltage from the source divides among the components.
To analyze one, add the resistances to get the equivalent resistance, . Then find the current with . Finally, find each voltage drop with for that resistor.
The bigger resistor gets the bigger share of the voltage. In a loop with one source and resistors, the voltage drops add up to the source voltage. This follows from Kirchhoff's loop rule and conservation of energy: energy supplied by the source is transferred to the resistors.
Because there's only one path, opening a switch or removing one bulb stops current everywhere. Adding resistors in series raises . If the source voltage stays fixed, the current decreases.
Worked example. A 12 V battery is in series with a 2 and a 4 resistor. First, . Next, A through both resistors. Then V and V. As a check, V, which matches the battery.
Check yourself: Three identical 5 bulbs are in series with a 15 V battery. What is the voltage across each bulb? (Answer: , so A, and each bulb gets V.)
In a parallel circuit, each component sits on its own branch between the same two junctions. Every branch gets the full source voltage, and the current splits among the branches.
Since each branch sees the same voltage, you can find each branch current on its own with . The lowest-resistance branch carries the most current. The total current from the source is the sum of the branch currents, which is Kirchhoff's junction rule.
For equivalent resistance, add the reciprocals, , then flip the answer. For two resistors you can use . The result is always smaller than the smallest branch, because each new branch gives charge another path.
Home circuits use parallel branches so one device switching off doesn't shut off the others. In the United States, many household outlets supply about 120 V, while some supply 240 V. For combination circuits, collapse the purely series or purely parallel groups one at a time until one resistor remains.
Worked example. A 12 V battery is connected across a 4 and a 6 resistor in parallel. The branch currents are A and A, so the total is 5 A. The equivalent resistance is , and A confirms it.
Check yourself: You add a third 12 resistor in parallel to the 4 and 6 pair on the 12 V battery. What is the new total current? (Answer: The new branch carries A, so the total rises from 5 A to 6 A. The other branch currents don't change.)
| Quantity | Series | Parallel |
|---|---|---|
| Current | Same through every component | Splits among branches, and the branch currents add to the total |
| Voltage | Splits among components, and the drops add to the source voltage | Same across every branch |
| Equivalent resistance | $R_1 + R_2 + \dots$, larger than any single resistor | $1/R_{eq} = 1/R_1 + 1/R_2 + \dots$, smaller than the smallest resistor |
| Adding another resistor | At fixed source voltage, total current goes down | At fixed source voltage, total current goes up |
| One bulb burns out | Every bulb goes dark | The other bulbs stay lit |
Electric power is the rate at which a circuit converts electrical energy into heat, light, or motion. It's measured in watts, where 1 W = 1 J/s.
For a direct-current circuit, power is . For a resistor, substituting gives and . Use the voltage across and current through that resistor. A source value works only if it is also the resistor's value. Average power in an alternating-current circuit can require an additional factor called the power factor.
In series, the current is shared, so shows the larger resistor dissipates more power. In parallel, the voltage is shared, so shows the smaller resistor dissipates more. Either way, the total power equals the sum of the individual powers.
For constant power, energy is power times time, . Utilities bill in kilowatt-hours, where 1 kWh = 1000 W for 1 hour = 3.6 million J.
Worked example. An ideal resistive heater uses 1500 W at 120 V. Its current is A, and its resistance is . If it runs for 10 minutes a day for 30 days, that's 5 hours, so kWh. At $0.15 per kWh, the energy cost is about $1.13.
Check yourself: A 4 and an 8 resistor are in series with a 12 V battery. Which one dissipates more power, and how much? (Answer: The current is A. The 8 resistor dissipates W, and the 4 resistor dissipates 4 W.)
| Term | Definition |
|---|---|
| Electric current | The rate at which charge flows past a point, $I = Q/t$, measured in amperes. |
| Potential difference | The energy per coulomb between two points in a circuit, measured in volts and also called voltage. |
| Resistance | A component's opposition to current, measured in ohms, which depends on its material, length, and thickness. |
| Ohm's law | The relationship $V = IR$ between voltage, current, and resistance for an ohmic component. |
| Equivalent resistance | The single resistance that draws the same total current from the source as a group of resistors. |
| Series circuit | A circuit with one path, where every component carries the same current. |
| Parallel circuit | A circuit with multiple branches, where every branch has the same voltage across it. |
| Electric power | The rate of electrical energy conversion, found with $P = VI$ and measured in watts. |
| Kilowatt-hour | A unit of energy equal to 1000 W used for one hour, or 3.6 million joules. |
| Short circuit | A very low-resistance path that lets a dangerously large current flow. |
The bulbs convert energy, not charge, so the current leaving a bulb equals the current entering it. In a series circuit, the current is the same everywhere. What drops across each bulb is voltage, which is energy per coulomb.
Adding gives , which is , not . Flip it to get 2.4 . A quick check is that the answer has to be smaller than the smallest branch resistor.
At a fixed source voltage, adding resistance in series lowers the current. Adding a branch in parallel lowers the equivalent resistance, so the total current from the source goes up. This is why plugging too many devices into one circuit can trip a breaker.
In a series circuit, a resistor only gets its share of the source voltage, so plugging 12 V into for one resistor overstates its power. Find that resistor's own voltage drop or current first, then use the power formula.