4.5 Ohm's law

Ohm's law describes the relationship between voltage, current, and resistance in electrical circuits. It's one of the most fundamental tools you'll use to analyze circuits, calculate unknowns, and predict how components behave. Nearly every topic in circuit analysis builds on this relationship, so getting comfortable with it now pays off throughout the course.

Ohm's law fundamentals

Definition of Ohm's law

Ohm's law states that the current through a conductor is directly proportional to the voltage across it, as long as the resistance stays constant. In plain terms: increase the voltage, and the current increases proportionally. Double the voltage across a resistor, and you double the current through it.

This relationship holds well for ohmic materials like metals and many standard resistors. It breaks down for non-ohmic materials like semiconductors and diodes, where the current-voltage relationship is nonlinear.

Mathematical expression

The law is expressed as:

$V = IR$

where V is voltage (in volts), I is current (in amperes), and R is resistance (in ohms).

You can rearrange this to solve for any of the three quantities:

• To find current: $I = \frac{V}{R}$
• To find resistance: $R = \frac{V}{I}$

Quick example: A 12 V battery drives current through a 4 Ω resistor. The current is $I = \frac{12}{4} = 3 \text{ A}$.

Units of measurement

• Voltage (V): measured in volts (V). One volt equals one joule per coulomb.
• Current (I): measured in amperes (A). One ampere equals one coulomb per second.
• Resistance (R): measured in ohms (Ω). One ohm equals one volt per ampere.

You'll frequently see metric prefixes: millivolts (mV), milliamps (mA), kilohms (kΩ), megohms (MΩ). Getting comfortable converting between these is important for problem-solving.

Components of Ohm's law

Electric current

Current is the flow of electric charge through a conductor. In metals, this means electrons are drifting through the material. In electrolytes, ions carry the charge instead.

One ampere equals one coulomb of charge passing a point per second. Current can be direct current (DC), which flows in one direction, or alternating current (AC), which periodically reverses direction. Current also produces magnetic fields around the conductor and generates heat as charges collide with atoms in the material.

Voltage

Voltage is the electric potential difference between two points in a circuit. Think of it as the "push" that drives charges through the circuit. Without a voltage difference, no current flows.

One volt equals one joule of energy per coulomb of charge. Voltage sources include batteries, generators, and solar cells. The voltage across a component determines how much energy each coulomb of charge transfers to that component.

Resistance

Resistance is a material's opposition to current flow. A higher resistance means less current for a given voltage.

One ohm is defined as the resistance that allows one ampere of current when one volt is applied. Resistance depends on:

• Material: copper has very low resistivity; rubber has extremely high resistivity
• Length: longer conductors have more resistance
• Cross-sectional area: thicker conductors have less resistance
• Temperature: resistance changes with temperature (more on this below)

Resistance converts electrical energy into heat through Joule heating, which is why resistors warm up during operation.

Applications of Ohm's law

Circuit analysis

Ohm's law lets you find any unknown quantity (V, I, or R) when you know the other two. In a simple circuit with one resistor and a battery, this is straightforward. In more complex circuits, you combine Ohm's law with series/parallel rules and Kirchhoff's laws to solve for multiple unknowns.

It's also useful for troubleshooting. If you measure an unexpected voltage drop across a component, Ohm's law helps you figure out whether the resistance has changed or the current is off from what you expected.

Power calculations

Power tells you how fast energy is being converted in a circuit element. Using Ohm's law, you can express power in three equivalent forms:

$P = VI = I^2R = \frac{V^2}{R}$

Example: A 100 Ω resistor carries 0.5 A of current. The power dissipated is $P = (0.5)^2 \times 100 = 25 \text{ W}$. That's 25 joules of heat generated every second.

Choose whichever form of the power equation uses the two quantities you already know.

Series vs parallel circuits

Series circuits connect components end-to-end so the same current flows through each one.

• Total resistance: $R_{total} = R_1 + R_2 + R_3 + \ldots$
• Voltage divides across components in proportion to their resistances
• If one component breaks the circuit, current stops everywhere

Parallel circuits connect components across the same two nodes so they share the same voltage.

• Total resistance: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$
• Current divides among branches inversely proportional to their resistances
• The total resistance is always less than the smallest individual resistance

Most real circuits are combinations of series and parallel connections. You simplify them step by step, reducing parallel groups and series chains until you find the equivalent resistance of the whole network.

Limitations of Ohm's law

Non-ohmic materials

Not all materials obey Ohm's law. Non-ohmic (or nonlinear) materials have current-voltage curves that aren't straight lines. Examples include:

• Diodes: conduct easily in one direction but block current in the other
• Varistors: resistance drops sharply above a threshold voltage
• Filament light bulbs: resistance increases as the filament heats up

These components are useful precisely because of their nonlinear behavior, but you can't just plug values into $V = IR$ and expect accurate results. They require more complex models.

Temperature effects

Resistance changes with temperature for most materials. In metals, resistance increases as temperature rises because atoms vibrate more and scatter electrons more frequently. In semiconductors, resistance typically decreases with temperature because more charge carriers become available.

This relationship is described by the temperature coefficient of resistance. For high-power applications or precision circuits, temperature effects can cause significant deviations from the simple Ohm's law prediction.

High-frequency limitations

Ohm's law in its basic form assumes steady-state DC conditions. At high frequencies, other effects come into play:

• Skin effect: current concentrates near the surface of a conductor, increasing effective resistance
• Parasitic capacitance and inductance: components that behave as pure resistors at DC start acting like capacitors or inductors at high frequencies

At very high frequencies (RF and microwave ranges), you need to use complex impedance instead of simple resistance, and transmission line theory replaces basic circuit analysis.

Experimental verification

Experimental setup

A typical Ohm's law experiment uses:

1. A variable DC power supply (to set different voltages)
2. A resistor of known value
3. An ammeter in series to measure current
4. A voltmeter in parallel to measure voltage across the resistor

Safety note: include a current-limiting resistor to protect components, and don't exceed the power rating of your resistor.

Data collection methods

1. Set the power supply to a low voltage
2. Record the voltage across the resistor (from the voltmeter) and the current through it (from the ammeter)
3. Increase the voltage in equal steps, recording V and I at each step
4. Collect enough data points (at least 6-8) to establish a clear trend
5. Repeat measurements to check reproducibility

Error analysis

Plot voltage vs. current. For an ohmic resistor, you should get a straight line through the origin. The slope of that line equals the resistance.

• Calculate uncertainties in your voltage and current readings (based on meter precision)
• Propagate those uncertainties to find the uncertainty in your calculated resistance
• Use error bars on your plot and perform a linear regression to find the best-fit slope
• Compare your experimental resistance to the labeled value and discuss any discrepancies (contact resistance, meter loading, heating effects)

Ohm's law in complex circuits

Kirchhoff's laws

For circuits with multiple loops or branches, Ohm's law alone isn't enough. You also need:

• Kirchhoff's Current Law (KCL): The total current entering any junction equals the total current leaving it. This is conservation of charge.
• Kirchhoff's Voltage Law (KVL): The sum of all voltage gains and drops around any closed loop is zero. This is conservation of energy.

You apply Ohm's law to express the voltage drop across each resistor ($V = IR$), then use KCL and KVL to set up a system of equations you can solve for the unknowns.

Equivalent resistance

Complex resistor networks can be reduced to a single equivalent resistance that draws the same current from the source.

• Series: $R_{eq} = R_1 + R_2 + R_3 + \ldots$
• Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$
• Delta-Wye (Δ-Y) transformations handle configurations that aren't purely series or parallel

The strategy is to identify series and parallel groups, simplify them, and repeat until the entire network reduces to one resistor.

Voltage dividers

A voltage divider is two resistors in series that produce an output voltage that's a fraction of the input voltage:

$V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2}$

Example: With $V_{in} = 10 \text{ V}$, $R_1 = 3 \text{ k}\Omega$, and $R_2 = 7 \text{ k}\Omega$, the output is $V_{out} = 10 \times \frac{7}{10} = 7 \text{ V}$.

Voltage dividers are used for biasing transistors, scaling measurement signals, and setting reference voltages. One important caveat: if you connect a load to the output, it draws current and changes the effective resistance, which shifts $V_{out}$. This is called the loading effect.

Historical context

Georg Ohm's discovery

Georg Ohm, a German physicist, formulated his law in the 1820s through careful experiments with wires of different lengths and materials. He published his results in 1827 in Die galvanische Kette, mathematisch bearbeitet (The Galvanic Circuit, Investigated Mathematically). His work initially met skepticism, but it eventually became one of the cornerstones of electrical science.

Evolution of electrical theory

Ohm's law built on earlier discoveries by Volta (the battery), Ampère (the relationship between current and magnetism), and Faraday (electromagnetic induction). It later influenced Kirchhoff's circuit laws and Maxwell's electromagnetic theory. The quantitative framework Ohm established made it possible to design and analyze electrical systems systematically, driving the rapid growth of electrical technology through the 19th and 20th centuries.

Practical implications

Electrical safety

Ohm's law helps you understand why electrical safety matters quantitatively. The current through your body depends on the voltage you contact and your body's resistance: $I = \frac{V}{R_{body}}$. Even moderate voltages can drive dangerous currents if resistance is low (wet skin, for example).

Circuit breakers and fuses are rated using Ohm's law principles to interrupt current before it reaches dangerous levels. Wire gauges are selected so that the resistance doesn't cause excessive heating at the expected current.

Energy efficiency

Power losses in wires and components follow $P = I^2R$. This is why power transmission lines use very high voltages: for a given amount of power delivered, higher voltage means lower current, which means lower $I^2R$ losses in the wires.

In device design, choosing lower-resistance conductors and optimizing component values reduces wasted energy and heat generation.

Circuit design considerations

Ohm's law guides everyday design decisions:

• Selecting a current-limiting resistor for an LED (you need to drop the excess voltage while allowing the right current)
• Choosing resistor values for voltage dividers and biasing networks
• Calculating heat dissipation to ensure components stay within their thermal ratings
• Sizing power supplies to deliver adequate current at the required voltage

Advanced concepts

Conductivity vs resistivity

Resistivity ($\rho$) is an intrinsic property of a material that quantifies how strongly it resists current. Conductivity ($\sigma$) is its inverse: $\sigma = \frac{1}{\rho}$.

Resistance depends on both the material and the geometry of the conductor:

$R = \frac{\rho L}{A}$

where $L$ is the length and $A$ is the cross-sectional area. This equation explains why a long, thin wire has more resistance than a short, thick one made of the same material.

Copper has a resistivity of about $1.7 \times 10^{-8} \text{ Ω·m}$, while glass is around $10^{10} \text{ Ω·m}$. That enormous range is why material selection matters so much in circuit design.

Microscopic interpretation

At the atomic level, the Drude model explains Ohm's law by treating conduction electrons as a gas of particles that accelerate under an applied electric field and then scatter off lattice atoms. The average drift velocity of the electrons is proportional to the applied field, which produces the linear V-I relationship.

Key ideas from this model:

• Drift velocity is surprisingly slow (on the order of mm/s), even though the electric field propagates at nearly the speed of light
• Electron mobility describes how easily electrons move through the material
• Scattering increases with temperature in metals, which is why resistance rises with temperature

The Drude model has limitations, but it provides a useful bridge between the macroscopic law ($V = IR$) and the microscopic physics of charge transport.

Ohm's law in semiconductors

Semiconductors like silicon don't follow Ohm's law across their full operating range. Their conductivity depends heavily on doping (adding impurities to control charge carrier concentration) and temperature.

In certain limited regions of operation, a semiconductor device may behave approximately ohmically. But in general, the current-voltage relationship is nonlinear. Understanding where and why Ohm's law breaks down in semiconductors is the starting point for studying diodes, transistors, and integrated circuits.