01 · Introductory

The resistor and Ohm's law

A linear resistor links the voltage across it to the current through it.

80–105 minIntroductory

In this lesson

State Ohm's law with an explicit sign convention.
Identify voltage and current on a resistor before calculation.
Compute any one of V, I or R from the other two.
Distinguish a constitutive law from a conservation law.

Starting question

If a resistor has 5 V across it, what determines the current through it?

For an ideal linear resistor, one parameter—its resistance R—links the voltage across the two terminals to the current through the element. The important word is model: Ohm's law is a constitutive relation for a particular class of behaviour, not a universal law that every two-terminal device must obey.

01.1First locate V and I on the element

Figure 1.1 — Voltage and current references on a resistorCurrent I enters the positive terminal of V_R: receiver convention. The AoN voltage arrow points from − to +.

The figure uses receiver convention: the positive current reference enters the terminal defined positive by V_R. With the AoN voltage arrow pointing from − toward +, the V and I arrows point in opposite directions.

For an ideal linear resistor under this convention,

V_R = R I

R is positive for an ordinary passive resistor. If we reverse either reference without changing the other, the algebraic form must be handled consistently; the physical resistor has not changed.

01.2Constitutive law versus conservation law

Ohm's law tells us how one element behaves: it connects that element's terminal voltage and current. This is a constitutive law.

Later, Kirchhoff's laws will tell us how currents and voltages are constrained by network connection. Those are not properties of one resistor. Keeping these categories separate is fundamental:

QuestionType of relationExample

How does this element relate V and I?constitutiveV=RI

How do connected branches balance?network/conservationKirchhoff's current law and Kirchhoff's voltage law later

A circuit solution usually needs both kinds of statement: element laws and connection laws.

01.3Three algebraic forms, one physical relation

V = RI

I = V/R

R = V/I

The unit follows directly:

1 Ω = 1 V/A

Example 1. A 1.0 kΩ resistor has 5.0 V across it:

I = 5.0 V / 1.0 kΩ = 5.0 mA

Example 2. A resistor carries 2.0 mA when 6.6 V is applied:

R = 6.6 V / 2.0 mA = 3.3 kΩ

01.4A negative result is compatible with Ohm's law

Suppose the receiver-oriented references are fixed and the measured resistor voltage is V=−2 V with R=1 kΩ. Then

I = V/R = −2 mA

The negative current means the actual conventional current is opposite to the chosen arrow. The resistor is still described by the same linear relation.

Sanity check — what is the absorbed power?

P=VI=(−2 V)(−2 mA)=+4 mW. Both signed variables reversed relative to the chosen references, while the passive resistor still absorbs positive power.

01.5Ohm's law is an idealized linear behaviour, not a universal truth

Feynman emphasizes an important point: for many ordinary conducting materials the current is approximately proportional to terminal voltage over a useful range, but real materials are not required to remain perfectly linear under every temperature, field or frequency condition.

The ideal resistor used in circuit theory is therefore an abstraction: R is constant and the V–I relation is a straight line through the origin. A real resistor can change with temperature, self-heating, voltage stress, frequency and construction.

01.6Practice for mastery

2.2 kΩ carries 3.0 mA. Find the resistor voltage.
12 V produces 0.40 mA. Find R.
A receiver-oriented resistor has V=−3 V and R=1.5 kΩ. Find signed I and interpret its direction.
Explain in one sentence why “every electrical device obeys Ohm's law” is false.
Classify V=RI as a constitutive relation or a network conservation relation, and explain why.

Solutions

1. 6.6 V.

2. 30 kΩ.

3. I=−2 mA, so 2 mA flows opposite to the current reference.

4. Ohm's law describes a linear resistive model; many devices have nonlinear or condition-dependent V–I relations.

5. Constitutive: it describes the behaviour of one resistor at its terminals.

The next chapter asks what the parameter R itself means, how conductance G provides the reciprocal viewpoint, and how geometry and temperature connect the ideal number to a real resistor.