Theory — Ohm's Law and Resistor Networks
Charge, Current, and Voltage
Electric current (I) is the rate of flow of electric charge past a point in a circuit, measured in amperes (A). Voltage (V), also called potential difference, is the energy delivered per unit charge between two points, measured in volts (V). A battery maintains a potential difference that drives current through a closed circuit.
Voltage: 1 V = 1 joule / coulomb
Resistance: 1 Ω = 1 V / 1 A
Ohm's Law
For an ohmic resistor, the current through the resistor is directly proportional to the voltage across it. The constant of proportionality is the resistance R:
Equivalent forms:
I = V / R R = V / I
Section I — Ohm's Law
Connect a single resistor to a variable battery. Vary the voltage and measure the current at each setting. Plot V vs I — the slope of the best-fit line is the resistance. Repeat for several resistors to verify V = IR.
Section II — Series and Parallel Networks
Combine two resistors in series, then in parallel. Measure the total current and the voltage across each resistor. Compare the measured values to the predicted equivalent resistance and verify Kirchhoff's voltage and current laws.
Equivalent Resistance — Series
Resistors in series carry the same current. The total voltage across the combination is the sum of the individual voltage drops, so the equivalent resistance adds:
V_total = V₁ + V₂ + …
Equivalent Resistance — Parallel
Resistors in parallel share the same voltage. The total current is the sum of the branch currents, so the reciprocals of the resistances add:
I_total = I₁ + I₂ + …
Kirchhoff's Laws
Voltage Law (Loop Rule)
Around any closed loop, the sum of the voltage rises (sources) equals the sum of the voltage drops (resistors). Energy is conserved — a charge returning to its starting point has gained and lost equal amounts of energy.
Current Law (Junction Rule)
At any junction, the total current flowing in equals the total current flowing out. Charge is conserved — charge cannot build up or disappear at a single point in a steady-state circuit.
Comparing Calculated and Experimental Values
In this lab you will always compute a resistance or voltage using the formula, then compare it to the experimental value you get from reading the simulated ammeter or voltmeter. Because the circuit is ideal (no meter loading, no wire resistance), the two values should agree exactly. If they disagree, re-check which formula you applied and whether you converted mA to A before dividing.
R_experimental = V₀ / I_total
Instructions — Running the Virtual Experiment
Section I — Verifying Ohm's Law
Section II — Series and Parallel Networks
Simulation — Circuit Builder
Section I Data Table — Ohm's Law
For each resistor, calculate the current expected from Ohm's Law at each voltage (I = V / R, in mA) and type it into the "I calc" columns. Then read the actual current from the ammeter in the simulator and type it into "I exp" for each voltage. Click Calculate Section I to see a ✓/✗ match mark in each experimental cell.
| Nominal R (Ω) | V = 3 V | V = 6 V | V = 9 V | V = 12 V | ||||
|---|---|---|---|---|---|---|---|---|
| I calc (mA) | I exp (mA) | I calc (mA) | I exp (mA) | I calc (mA) | I exp (mA) | I calc (mA) | I exp (mA) | |
| 100 | ||||||||
| 220 | ||||||||
| 470 | ||||||||
| 1000 | ||||||||
| 2200 | ||||||||
Section II Data Table — Series & Parallel (V₀ = 9 V)
Enter the total current you read from the ammeter. Then type your calculated R_eq (series: R₁+R₂; parallel: R₁R₂/(R₁+R₂)), V₁ across R₁, and V₂ across R₂. In series the same current flows through both, so V₁ = I·R₁ and V₂ = I·R₂. In parallel both resistors share the full battery voltage, so V₁ = V₂ = V₀. Click Calculate Section II to reveal the experimental values and the match check.
| Config | R₁ (Ω) | R₂ (Ω) | I_total (mA) | R_eq calc (Ω) | R_eq exp (Ω) | V₁ calc (V) | V₁ exp (V) | V₂ calc (V) | V₂ exp (V) | Match? |
|---|---|---|---|---|---|---|---|---|---|---|
| Series | 100 | 220 | — | — | — | — | ||||
| Series | 220 | 470 | — | — | — | — | ||||
| Series | 470 | 1000 | — | — | — | — | ||||
| Parallel | 100 | 220 | — | — | — | — | ||||
| Parallel | 220 | 470 | — | — | — | — | ||||
| Parallel | 470 | 1000 | — | — | — | — |
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission.
Voltage, Resistance and Current: Ohm's Law and Resistor Networks
Physics 171 | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To verify Ohm's Law (V = IR) for five resistors by measuring the current at four different voltages and checking that V/I is constant for each resistor. The lab then tests the rules for equivalent resistance in series (R_eq = R₁ + R₂) and parallel (R_eq = R₁R₂ / (R₁ + R₂)) combinations, and confirms Kirchhoff's voltage and current laws for each configuration.
Theory
Ohm's Law states that the current through an ohmic resistor is directly proportional to the voltage across it:
V = I · R
A graph of V versus I for an ohmic resistor is a straight line through the origin with slope equal to the resistance. The electrical power dissipated by the resistor is P = VI = I²R = V²/R.
For resistors in series, the same current flows through each resistor and the individual voltage drops add to the source voltage (Kirchhoff's voltage law):
R_series = R₁ + R₂ V_total = V₁ + V₂
For resistors in parallel, the same voltage appears across each resistor and the branch currents add to the total current (Kirchhoff's current law):
1 / R_parallel = 1/R₁ + 1/R₂ I_total = I₁ + I₂
Because the circuit in this simulation is ideal, the calculated value of each quantity (R_eq, V₁, I_total) should match the experimental value read from the ammeter or voltmeter exactly. Any disagreement means the wrong formula was applied, the wrong configuration was assumed (series vs parallel), or units were not converted correctly.
Calculations — Sample: R = 470 Ω at V = 9 V (Section I)
Current from Ohm's Law: I = V / R = 9.00 V / 470 Ω = 0.01915 A = 19.15 mA
Power dissipated: P = VI = (9.00)(0.01915) = 0.1723 W
Measured resistance from slope of V-vs-I graph: R = 9.00 V / 0.01915 A = 470.0 Ω (matches nominal 470 Ω)
Series sample — R₁ = 220 Ω, R₂ = 470 Ω, V₀ = 9 V:
R_series = 220 + 470 = 690 Ω
I_total = 9.00 / 690 = 0.01304 A = 13.04 mA
V₁ = I × R₁ = (0.01304)(220) = 2.87 V
V₂ = I × R₂ = (0.01304)(470) = 6.13 V
Check: V₁ + V₂ = 2.87 + 6.13 = 9.00 V ✓ (Kirchhoff's voltage law)
Parallel sample — R₁ = 220 Ω, R₂ = 470 Ω, V₀ = 9 V:
R_parallel = (220 × 470) / (220 + 470) = 103 400 / 690 = 149.86 Ω
I_total = 9.00 / 149.86 = 0.06006 A = 60.06 mA
I₁ = V / R₁ = 9.00 / 220 = 0.04091 A = 40.91 mA
I₂ = V / R₂ = 9.00 / 470 = 0.01915 A = 19.15 mA
Check: I₁ + I₂ = 40.91 + 19.15 = 60.06 mA ✓ (Kirchhoff's current law)
Results Table
Section I — Ohm's Law (currents in mA; calculated from I = V/R, experimental read from the ammeter)
| Nominal R (Ω) | V = 3 V | V = 6 V | V = 9 V | V = 12 V | ||||
|---|---|---|---|---|---|---|---|---|
| I calc | I exp | I calc | I exp | I calc | I exp | I calc | I exp | |
| 100 | 30.00 | 30.00 ✓ | 60.00 | 60.00 ✓ | 90.00 | 90.00 ✓ | 120.00 | 120.00 ✓ |
| 220 | 13.64 | 13.64 ✓ | 27.27 | 27.27 ✓ | 40.91 | 40.91 ✓ | 54.55 | 54.55 ✓ |
| 470 | 6.38 | 6.38 ✓ | 12.77 | 12.77 ✓ | 19.15 | 19.15 ✓ | 25.53 | 25.53 ✓ |
| 1000 | 3.00 | 3.00 ✓ | 6.00 | 6.00 ✓ | 9.00 | 9.00 ✓ | 12.00 | 12.00 ✓ |
| 2200 | 1.36 | 1.36 ✓ | 2.73 | 2.73 ✓ | 4.09 | 4.09 ✓ | 5.45 | 5.45 ✓ |
Section II — Series and Parallel Combinations (V₀ = 9 V)
| Config | R₁ (Ω) | R₂ (Ω) | I_total (mA) | R_eq calc (Ω) | R_eq exp (Ω) | V₁ calc (V) | V₁ exp (V) | V₂ calc (V) | V₂ exp (V) | Match? |
|---|---|---|---|---|---|---|---|---|---|---|
| Series | 100 | 220 | 28.13 | 320.0 | 320.0 | 2.81 | 2.81 | 6.19 | 6.19 | R ✓ · V₁ ✓ · V₂ ✓ |
| Series | 220 | 470 | 13.04 | 690.0 | 690.0 | 2.87 | 2.87 | 6.13 | 6.13 | R ✓ · V₁ ✓ · V₂ ✓ |
| Series | 470 | 1000 | 6.12 | 1470.0 | 1470.0 | 2.88 | 2.88 | 6.12 | 6.12 | R ✓ · V₁ ✓ · V₂ ✓ |
| Parallel | 100 | 220 | 130.91 | 68.75 | 68.75 | 9.00 | 9.00 | 9.00 | 9.00 | R ✓ · V₁ ✓ · V₂ ✓ |
| Parallel | 220 | 470 | 60.06 | 149.86 | 149.86 | 9.00 | 9.00 | 9.00 | 9.00 | R ✓ · V₁ ✓ · V₂ ✓ |
| Parallel | 470 | 1000 | 28.15 | 319.73 | 319.73 | 9.00 | 9.00 | 9.00 | 9.00 | R ✓ · V₁ ✓ · V₂ ✓ |
Note: in series, V₁ + V₂ = V₀ (Kirchhoff). In parallel, V₁ = V₂ = V₀ because the resistors share the full battery voltage.
Discussion
In Section I the measured current was perfectly proportional to the applied voltage for every resistor, confirming Ohm's Law. Doubling the voltage from 3 V to 6 V exactly doubled the current; tripling the voltage to 9 V tripled it; quadrupling to 12 V quadrupled it. V / I was constant at every row within each resistor and matched the nominal resistance to three significant figures, so the V-vs-I graph for each resistor was a straight line through the origin with the correct slope. This confirms that the carbon-film resistors used in the simulation are ohmic over the voltage range tested.
In Section II the series combinations gave equivalent resistances equal to the sum of the individual resistances, and the parallel combinations gave the expected (R₁·R₂)/(R₁+R₂) values. The total current measured by the ammeter agreed with V₀/R_eq for every pair. For the representative case R₁ = 220 Ω and R₂ = 470 Ω in series, the individual voltage drops summed to exactly the battery voltage (2.87 V + 6.13 V = 9.00 V), confirming Kirchhoff's voltage law. For the same pair in parallel, the individual branch currents summed to the total current (40.91 mA + 19.15 mA = 60.06 mA), confirming Kirchhoff's current law.
The key experimental pattern is that series combinations always give a larger equivalent resistance than either component, while parallel combinations always give a smaller equivalent resistance than either component. This is why adding resistors in series reduces the current (the battery has to push charge through more resistance), while adding resistors in parallel increases the total current (the battery has additional paths through which charge can flow).
Conclusion
The experiment successfully verified Ohm's Law and the rules for combining resistors. Every resistor produced a constant V/I ratio matching its nominal resistance, so the simulated resistors are ohmic. The series rule R_eq = R₁ + R₂ and the parallel rule R_eq = R₁R₂/(R₁+R₂) were confirmed for three independent pairs in each configuration. Kirchhoff's voltage law held exactly in every series circuit and Kirchhoff's current law held exactly in every parallel circuit. Together these results confirm that the equivalent-resistance rules are direct consequences of the conservation of energy (KVL) and the conservation of charge (KCL) applied to resistor networks.
Practice Questions
Show all work and include units in your answers.