Unit 15 - Grade 11-12 Physics

Current Electricity and Circuits

Study electric current, voltage, resistance, Ohm's law, resistivity, electric power, series and parallel circuits, Kirchhoff's rules, meters, capacitors, household electricity, and RC circuit behavior.

Lesson roadmap

What Students Should Master in This Unit

Current electricity explains how charges move through circuits, how energy is transferred by batteries and power supplies, and how circuit components control current, voltage, resistance, and power.

Model moving charge

Connect current, charge flow, voltage, resistance, drift speed, and conventional current direction.

Analyze complete circuits

Use Ohm's law, equivalent resistance, series/parallel rules, meters, and Kirchhoff's rules.

Calculate power and energy

Solve electric power, energy use, battery energy, capacitors, household electricity, and RC circuit problems.

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Current Basics Voltage and Energy Resistance and Resistivity Ohm's Law Circuit Diagrams Meters and Measurements Series Circuits Parallel Circuits Mixed Circuits Kirchhoff's Rules Power and Energy Household Electricity Capacitors RC Circuits Simulation Labs Lab Skills Worked Examples Practice Problems
Moving electric charge

1. Electric Current Basics

Electric current is the rate at which charge passes a point in a circuit. In metal wires, electrons drift through the conductor, while conventional current is defined as the direction positive charge would move.

Closed circuit: conventional current vs electron drift battery + - closed switch R - conventional current electron drift in metal wires is opposite
Use conventional current direction for circuit diagrams, even though electrons drift the opposite way inside metal conductors.
Current definition I = ΔQ/Δt Unit: ampere, where 1 A = 1 C/s.
Charge moved ΔQ = IΔt Useful when current is steady over time.
Number of electrons N = |Q|/e e = 1.60 × 10-19 C.

Key Current Ideas

  • Conventional current: drawn from positive terminal toward negative terminal outside a battery.
  • Electron flow: actual electron drift in metals is opposite conventional current.
  • Closed circuit: current requires a complete conducting path.
  • Open circuit: a break stops continuous current.
  • Short circuit: a very low-resistance path that can create dangerously large current.
Energy per unit charge

2. Voltage, EMF, and Electric Energy

Voltage is electric potential difference. It tells how much energy is transferred per coulomb of charge moving between two points.

Voltage is measured across two points 12 V source resistor V ΔV = ΔU/q current flows through; voltage is measured across
A battery gives charges electric potential energy; circuit components convert that energy into useful forms.
Potential difference ΔV = ΔU/q Unit: volt, where 1 V = 1 J/C.
Energy transfer ΔU = qΔV A 12 V battery gives 12 J per coulomb ideally.
Battery EMF ε = W/q EMF is the ideal energy supplied per unit charge.

How to Think About Voltage

  • A battery raises electric potential energy of charges.
  • Resistors and devices convert electric energy into thermal, light, sound, or mechanical energy.
  • Voltage is measured between two points, not through one point.
  • Current flows through components; voltage is across components.
Opposition to current

3. Resistance and Resistivity

Resistance describes how strongly a component opposes current. Resistivity is a material property that helps determine the resistance of a wire or conductor.

Wire resistance: R = ρL/A long L increases R thin wire: smaller A larger A lowers R thicker wire: larger A material sets ρ
For a uniform wire, resistance increases with length and resistivity, but decreases when cross-sectional area is larger.
Resistance definition R = ΔV/I Unit: ohm, written Ω.
Wire resistance R = ρL/A Longer wires have larger resistance; wider wires have smaller resistance.
Temperature effect R = R0[1 + αΔT] For many metals, resistance increases as temperature rises.

Factors Affecting Resistance

  • Length: longer conductor means more resistance.
  • Area: larger cross-sectional area means less resistance.
  • Material: copper has low resistivity; nichrome has higher resistivity.
  • Temperature: metal resistance usually increases with temperature.
Voltage-current relationship

4. Ohm's Law

Ohm's law relates voltage, current, and resistance for ohmic materials or components at constant temperature.

Ohm's law: circuit measurement plus graph R A V V I slope = R V = IR
An ideal ohmic resistor gives a straight-line voltage-current graph; the slope of a V vs I graph is resistance.
Ohm's law ΔV = IR Voltage across a resistor equals current times resistance.
Current form I = ΔV/R Increasing voltage increases current if R is constant.
Resistance from graph R = slope of V vs I Ohmic resistor gives a straight-line V-I graph.
Common mistake: Ohm's law applies to ideal resistors and ohmic materials under constant conditions. Light bulbs, diodes, and some devices can be non-ohmic.
Visual circuit models

5. Circuit Diagrams

Schematic diagrams show how components are connected, not their physical location. These visuals help students identify series paths, parallel branches, meters, and current direction.

Closed circuit schematic battery closed switch lamp resistor
Simple Closed CircuitA battery supplies energy, current moves through a complete path, and components transform electric energy.
Series Parallel R1 R2 one path: same current R1 R2 branches: same voltage
Series vs ParallelSeries circuits have one current path; parallel circuits have branches with the same voltage across each branch.
Correct meter placement A ammeter in series resistor V voltmeter in parallel
Using MetersAn ammeter measures current through a component, while a voltmeter measures potential difference across a component.
Measuring circuit quantities

6. Meters and Measurements

Correct meter placement matters. A wrong connection can give incorrect readings or damage equipment.

Meter placement: current through, voltage across A series path resistor V parallel branch Wrong: ammeter across the source can create a near short circuit.
An ammeter must be placed in series so circuit current passes through it; a voltmeter is placed in parallel across the component.
Meter Measures Connection Rule Ideal Resistance
AmmeterCurrentConnect in series with the component.Nearly 0 Ω.
VoltmeterPotential differenceConnect in parallel across the component.Very large resistance.
OhmmeterResistanceConnect across an isolated component with power off.Uses its own internal battery.
MultimeterCurrent, voltage, or resistanceUse correct setting and ports before measuring.Depends on mode.
Lab warning: Never place an ammeter directly across a battery or power supply. That creates a near short circuit.
One path for current

7. Series Circuits

In a series circuit, components are connected end-to-end so there is only one path for current.

Series circuit: one path, same current everywhere R1 R2 Itotal = I1 = I2; Req = R1 + R2
Series components share the same current because there is only one continuous path for charge flow.
Same current Itotal = I1 = I2 = I3 Current is the same through every series component.
Voltage adds Vtotal = V1 + V2 + V3 Battery voltage is split among series components.
Equivalent resistance Req = R1 + R2 + R3 Adding series resistors increases total resistance.

Series Circuit Behavior

  • If one component breaks, the entire circuit opens.
  • The largest resistor gets the largest voltage drop.
  • Adding a resistor in series increases total resistance and lowers total current.
Multiple paths for current

8. Parallel Circuits

In a parallel circuit, components are connected across the same two points, creating separate branches.

Parallel circuit: each branch has the source voltage R1 R2 source V1 = V2 = Vsource Itotal = I1 + I2
Parallel branches have the same voltage, while total current divides among the available paths.
Same voltage Vtotal = V1 = V2 = V3 Each branch has the same voltage as the source, ideally.
Current adds Itotal = I1 + I2 + I3 Total current splits into branches.
Equivalent resistance 1/Req = 1/R1 + 1/R2 + 1/R3 Req is less than the smallest branch resistance.

Parallel Circuit Behavior

  • If one branch opens, other branches can still operate.
  • Adding a resistor in parallel decreases equivalent resistance.
  • The branch with lower resistance gets larger current.
Combining circuit rules

9. Mixed Series-Parallel Circuits

Many circuits contain both series and parallel parts. The safest strategy is to simplify one section at a time.

Mixed circuit: reduce the parallel block first R1 R2 R3 R23 replace R2 || R3, then add R1
Reduce one parallel or series group at a time, then work backward to find individual voltages and currents.
Reduce parallel group 1/Rp = Σ(1/R) Find the equivalent resistance of each branch group.
Then add series parts Rtotal = Rseries + Rp Combine simplified sections carefully.
Work backward V = IR After finding total current, reopen the circuit to find branch values.

Mixed-Circuit Strategy

  1. Redraw the circuit cleanly.
  2. Identify series-only and parallel-only groups.
  3. Replace one group with its equivalent resistance.
  4. Repeat until one total resistance remains.
  5. Find total current from the source voltage.
  6. Work backward to find individual branch currents and voltage drops.
Conservation rules for circuits

10. Kirchhoff's Rules

Kirchhoff's rules are conservation laws for circuits. They are especially useful for multi-loop circuits that cannot be solved by simple series-parallel reduction.

Kirchhoff rules: junction and loop conservation I1 in I2 I3 I1 = I2 + I3 ΣΔV = 0 around loop
At junctions, charge is conserved; around closed loops, the total voltage rise and drop must add to zero.
Junction rule ΣIin = ΣIout Conservation of charge at a junction.
Loop rule ΣΔV = 0 Conservation of energy around a closed loop.
Resistor drop ΔVR = -IR Moving with current through a resistor is a voltage drop.

Sign Convention Tips

  • Across a battery from negative to positive terminal: voltage rise.
  • Across a resistor in the direction of current: voltage drop.
  • If your final current is negative, the real current direction is opposite your chosen arrow.
Rate of energy transfer

11. Electric Power and Energy

Electric power is the rate at which electrical energy is transferred or converted into another form.

Power: the load converts electrical energy each second load light/heat P = IV Use P = I2R or P = V2/R for resistors.
Power is energy per time; in circuits, P = IV connects the device voltage and current to the rate of energy transfer.
Power definition P = E/t Unit: watt, where 1 W = 1 J/s.
Electrical power P = IV Works for any circuit element using current and voltage across it.
Resistor power P = I2R = V2/R Use Ohm's law to choose the convenient form.

Energy Use

  • Energy in joules: E = Pt.
  • Energy in kilowatt-hours: energy = power in kW × time in h.
  • Utility bills often charge per kilowatt-hour.
Real-world circuit safety

12. Household Electricity and Safety

Household circuits are usually wired in parallel so devices receive the same supply voltage and can operate independently.

Household model: protected parallel branches panel breaker lamp outlet/load grounded branch
Household branches operate in parallel and are protected by breakers, fuses, grounding, and GFCI devices.
Feature Purpose Physics Idea
Parallel wiringDevices operate independently.Each branch receives the supply voltage.
FuseMelts when current is too large.Breaks circuit to prevent overheating.
Circuit breakerSwitches off during overload.Protects wires from excessive current.
Ground wireProvides a safer low-resistance path.Reduces shock risk during faults.
GFCI outletDetects current imbalance.Opens circuit quickly if leakage occurs.
Professional note: Classroom circuit calculations use simplified models. Real household wiring should only be handled by qualified adults following local electrical codes.
Storing charge and energy

13. Capacitors

A capacitor stores separated charge on two conductors and stores energy in the electric field between them. Capacitors appear in camera flashes, timing circuits, filters, sensors, power supplies, and touch screens.

Capacitor symbol and plate model +Q -Q circuit symbol electric field C = ε0A/d
A capacitor separates equal and opposite charges on two conductors; the electric field between them stores energy.
Capacitance C = Q/ΔV Unit: farad, where 1 F = 1 C/V.
Charge on plates Q = CΔV The two plates hold equal magnitude charges, +Q and -Q.
Parallel-plate capacitor C = ε0A/d Larger plate area increases C; larger spacing decreases C.
Energy stored U = 1/2 C(ΔV)2 Also U = 1/2 QΔV = Q2/(2C).
Parallel capacitors Ceq = C1 + C2 + C3 Parallel capacitors share the same voltage.
Series capacitors 1/Ceq = 1/C1 + 1/C2 + 1/C3 Series capacitors carry the same charge magnitude.

What Capacitors Do in Circuits

  • Store charge separation and electric potential energy.
  • Block steady DC current after they are fully charged.
  • Temporarily allow current while charging or discharging.
  • Smooth voltage changes in power supplies.
  • Create timing behavior when paired with resistors in RC circuits.

Series vs Parallel Capacitors

Connection Same Quantity Addition Rule Key Idea
ParallelVoltageCapacitances add directly.More plate area is effectively available.
SeriesCharge magnitudeReciprocals add.Equivalent capacitance is smaller than the smallest capacitor.
Lab warning: Real capacitors can hold charge after power is removed. Use only low-voltage classroom capacitors and discharge them safely under teacher direction.
Circuits with capacitors

14. RC Circuits

An RC circuit contains a resistor and capacitor. These circuits are used for timing, filtering, sensors, camera flashes, and smoothing power supplies.

RC circuit: resistor controls capacitor charging speed R C τ VC 63% time
After one time constant, a charging capacitor reaches about 63 percent of its final voltage.
Time constant τ = RC After one time constant, charging reaches about 63% of final voltage.
Charging capacitor VC = V0(1 - e-t/RC) Capacitor voltage rises toward supply voltage.
Discharging capacitor VC = V0e-t/RC Capacitor voltage falls exponentially.

Conceptual RC Behavior

  • At the first instant of charging, an uncharged capacitor behaves like a wire.
  • After a long time in DC steady state, a fully charged capacitor behaves like an open circuit.
  • Larger R or C means a slower charge/discharge process.
Simulation labs

15. Simulation Labs for This Unit

These PhET simulations help students build circuits, test Ohm's law, compare series and parallel behavior, and connect circuit models to energy transfer.

Simulation lab: build, measure, then explain predict I, V, R A build and measure explain compare model
Students should treat a simulation as an investigation: make a prediction, build the circuit, collect readings, and explain the result.
Circuit Construction Kit: DC

Build circuits with batteries, bulbs, switches, wires, and meters to compare current, voltage, resistance, and brightness.

Lab idea: build one series circuit and one parallel circuit, then compare bulb brightness and current.
Open Simulation
Circuit Construction Kit: DC - Virtual Lab

Use a lab-focused circuit builder for more detailed measurements, circuit design, and troubleshooting tasks.

Lab idea: design a circuit where one bulb stays lit after another branch opens.
Open Simulation
Ohm's Law

Adjust voltage and resistance to see how current changes and reinforce V = IR relationships.

Lab idea: keep resistance fixed, double voltage, and record the current change.
Open Simulation
Resistance in a Wire

Explore how length, area, and resistivity affect wire resistance.

Lab idea: double wire length and explain the change using R = ρL/A.
Open Simulation
Battery-Resistor Circuit

Investigate voltage, current, resistance, and microscopic charge motion in a simple battery-resistor circuit.

Lab idea: compare electron motion with conventional current direction.
Open Simulation
Capacitor Lab: Basics

Explore capacitance, plate spacing, stored energy, and simple RC behavior.

Lab idea: change plate separation and observe how capacitance changes.
Open Simulation
Investigation skills

16. Current Electricity Lab Skills

Circuit labs reward careful setup, clean diagrams, and safe meter use. Students should always predict readings before measuring.

Lab setup: low voltage, power off before changes supply low voltage test resistor A/V predict wire safely measure compare
Strong lab work includes a clear diagram, safe low-voltage equipment, correct meter ports, and measured values compared with calculations.

Common Labs

  • Ohm's law lab using a resistor, variable voltage, ammeter, and voltmeter.
  • Series and parallel resistor comparison lab.
  • Equivalent resistance lab using multimeters.
  • Power and brightness lab with bulbs or resistors.
  • Resistivity lab using wires with different lengths and thicknesses.
  • Kirchhoff's junction and loop rule verification.
  • Capacitor charging and discharging lab using safe low-voltage equipment.

Good Data Habits

  • Turn power off before changing circuit wiring.
  • Check meter setting and ports before every measurement.
  • Use consistent units: amps, volts, ohms, watts, joules, seconds.
  • Record a circuit diagram next to every data table.
  • Compare measured equivalent resistance with calculated equivalent resistance.
Safety note: Use only teacher-approved low-voltage supplies for student circuits. Never experiment with wall outlets or household wiring.
Worked examples

17. Worked Examples

Example 1: Current from charge flow

12 C of charge passes a point in 4.0 s. Find current.

I = ΔQ/Δt = 12/4.0 = 3.0 A.

Example 2: Charge moved by current

A 2.5 A current flows for 8.0 s. How much charge passes?

ΔQ = IΔt = (2.5)(8.0) = 20 C.

Example 3: Ohm's law

A 6.0 Ω resistor is connected to 12 V. Find current.

I = V/R = 12/6.0 = 2.0 A.

Example 4: Resistance from wire dimensions

A wire has ρ = 1.7 × 10-8 Ω m, length 2.0 m, and area 1.0 × 10-6 m2. Find R.

R = ρL/A = (1.7 × 10-8)(2.0)/(1.0 × 10-6) = 0.034 Ω.

Example 5: Series resistance

Three resistors 2.0 Ω, 4.0 Ω, and 6.0 Ω are in series. Find Req.

Req = 2.0 + 4.0 + 6.0 = 12.0 Ω.

Example 6: Series current

The series circuit in example 5 is connected to 24 V. Find total current.

I = V/Req = 24/12.0 = 2.0 A.

Example 7: Series voltage drops

Using example 6, find voltage across the 6.0 Ω resistor.

V = IR = (2.0)(6.0) = 12 V.

Example 8: Parallel resistance

Find Req for 6.0 Ω and 3.0 Ω in parallel.

1/Req = 1/6.0 + 1/3.0 = 0.500, so Req = 2.0 Ω.

Example 9: Parallel branch currents

A 12 V battery is connected across 6.0 Ω and 3.0 Ω parallel branches. Find each branch current.

I6 = 12/6.0 = 2.0 A. I3 = 12/3.0 = 4.0 A.

Total current is 6.0 A.

Example 10: Electric power

A device uses 2.0 A at 120 V. Find power.

P = IV = (2.0)(120) = 240 W.

Example 11: Energy cost

A 1.5 kW heater runs for 4.0 h. Find energy use in kWh.

E = Pt = (1.5 kW)(4.0 h) = 6.0 kWh.

Example 12: Kirchhoff junction

5.0 A enters a junction. One branch carries 1.5 A out and another carries 2.0 A out. Find the third outgoing current.

5.0 = 1.5 + 2.0 + I3, so I3 = 1.5 A.

Example 13: Capacitor charge

A 220 µF capacitor is connected across 9.0 V. Find the charge stored on each plate.

Q = CΔV = (220 × 10-6)(9.0) = 1.98 × 10-3 C.

Each plate has equal magnitude charge: one plate is positive and the other is negative.

Example 14: Capacitor energy

A 10 µF capacitor is charged to 12 V. Find the stored energy.

U = 1/2 C(ΔV)2 = 0.5(10 × 10-6)(12)2 = 7.2 × 10-4 J.

Example 15: Capacitors in parallel

Find equivalent capacitance for 4.0 µF and 6.0 µF in parallel.

Ceq = C1 + C2 = 4.0 + 6.0 = 10.0 µF.

Example 16: RC time constant

A circuit has R = 100 kΩ and C = 20 µF. Find time constant.

τ = RC = (100 × 103)(20 × 10-6) = 2.0 s.

Example 17: Resistor power forms

A 10 Ω resistor carries 3.0 A. Find power.

P = I2R = (3.0)2(10) = 90 W.

Independent practice

18. Practice Problems

Try each problem first. Then open the answer check and compare circuit rules, units, voltage drops, branch currents, and power formulas.

1. 18 C of charge passes a point in 6.0 s. Find current.

Answer

I = Q/t = 18/6.0 = 3.0 A.

2. A 4.0 A current flows for 12 s. Find charge moved.

Answer

Q = It = (4.0)(12) = 48 C.

3. A resistor has 9.0 V across it and 0.30 A through it. Find R.

Answer

R = V/I = 9.0/0.30 = 30 Ω.

4. A 15 Ω resistor is connected to 45 V. Find current.

Answer

I = V/R = 45/15 = 3.0 A.

5. If voltage doubles and resistance stays constant, what happens to current?

Answer

Current doubles.

6. If resistance doubles and voltage stays constant, what happens to current?

Answer

Current is cut in half.

7. Find Req for 5 Ω, 10 Ω, and 15 Ω in series.

Answer

Req = 5 + 10 + 15 = 30 Ω.

8. A 30 Ω series circuit is connected to 120 V. Find current.

Answer

I = 120/30 = 4.0 A.

9. A 4.0 A current flows through a 10 Ω series resistor. Find its voltage drop.

Answer

V = IR = (4.0)(10) = 40 V.

10. Find Req for 4 Ω and 12 Ω in parallel.

Answer

1/Req = 1/4 + 1/12 = 1/3, so Req = 3 Ω.

11. A 24 V source is connected to 8 Ω and 12 Ω parallel branches. Find branch currents.

Answer

I8 = 24/8 = 3.0 A; I12 = 24/12 = 2.0 A.

12. Using problem 11, find total current.

Answer

Itotal = 3.0 + 2.0 = 5.0 A.

13. What happens to total resistance when another resistor is added in series?

Answer

Total resistance increases.

14. What happens to total resistance when another resistor is added in parallel?

Answer

Total resistance decreases.

15. A lamp uses 0.50 A at 120 V. Find power.

Answer

P = IV = (0.50)(120) = 60 W.

16. A 20 Ω resistor has 5.0 A through it. Find power.

Answer

P = I2R = (5.0)2(20) = 500 W.

17. A 100 W bulb runs for 10 h. Find energy use in kWh.

Answer

100 W = 0.100 kW, so E = (0.100)(10) = 1.0 kWh.

18. Three currents enter a junction: 2 A, 3 A, and 4 A. One current leaves. Find it.

Answer

Iout = 2 + 3 + 4 = 9 A.

19. In a closed loop, a 12 V battery is connected to one resistor. What is the resistor voltage drop?

Answer

12 V, assuming ideal wires and battery.

20. Where should a voltmeter be connected to measure voltage across a resistor?

Answer

In parallel across the resistor.

21. Where should an ammeter be connected to measure current through a resistor?

Answer

In series with the resistor.

22. A wire's length doubles while material and area stay the same. What happens to resistance?

Answer

Resistance doubles because R = ρL/A.

23. A 470 µF capacitor is connected to 6.0 V. Find the charge stored.

Answer

Q = CΔV = (470 × 10-6)(6.0) = 2.82 × 10-3 C.

24. A 5.0 µF capacitor is charged to 20 V. Find the stored energy.

Answer

U = 1/2 C(ΔV)2 = 0.5(5.0 × 10-6)(20)2 = 1.0 × 10-3 J.

25. Find equivalent capacitance for 3.0 µF and 9.0 µF in parallel.

Answer

Ceq = 3.0 + 9.0 = 12.0 µF.

26. Find equivalent capacitance for 3.0 µF and 6.0 µF in series.

Answer

1/Ceq = 1/3.0 + 1/6.0 = 0.500, so Ceq = 2.0 µF.

27. A circuit has R = 50 kΩ and C = 10 µF. Find time constant.

Answer

τ = RC = (50 × 103)(10 × 10-6) = 0.50 s.

28. In the Circuit Construction Kit simulation, why can one bulb stay lit when another parallel branch is removed?

Answer

Parallel branches provide separate current paths, so one open branch does not necessarily open the others.

Final review

19. What to Know Before Moving On

  • Electric current is I = ΔQ/Δt.
  • Voltage is energy per unit charge: ΔV = ΔU/q.
  • Resistance is R = V/I, and wire resistance is R = ρL/A.
  • Ohm's law is V = IR for ohmic components at constant temperature.
  • In series circuits, current is the same through every component.
  • In series circuits, voltages add and resistances add.
  • In parallel circuits, voltage is the same across each branch.
  • In parallel circuits, currents add and equivalent resistance follows 1/Req = Σ(1/R).
  • Kirchhoff's junction rule conserves charge: current in equals current out.
  • Kirchhoff's loop rule conserves energy: total voltage change around a closed loop is zero.
  • Electric power can be found using P = IV, P = I2R, or P = V2/R.
  • Energy use is E = Pt, often measured in kilowatt-hours for household electricity.
  • An ammeter goes in series; a voltmeter goes in parallel.
  • Capacitance is C = Q/ΔV, and capacitor energy is U = 1/2 C(ΔV)2.
  • Parallel capacitors add directly; series capacitors add by reciprocals.
  • RC circuit timing depends on τ = RC.
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