Electrical Resistance and Ohm's Law

E03: Electrical Resistance and Ohm’s Law

Foundations of Electricity and Electrical Measurement, mini-lecture 3.

E01 introduced voltage as a potential difference, and E02 identified electric current with the rate of transported charge. When a voltage is placed across a piece of conducting material, what determines the resulting current?

A Conductor does not respond only to the applied voltage. Its material, shape, and temperature also matter. Those effects are described by resistivity and resistance. For a broad class of materials under well-controlled conditions, the current is proportional to the voltage. This is Ohm’s law.

The circuit-level equation is

(1)

The corresponding local material relationship is

(2)

The goal of this lecture is not just to use the first equation. We will derive its connection to the second, examine where both relationships apply, and explain how to measure resistance in a real laboratory.

Learning objectives

1. distinguish resistance of a component from resistivity of a material;
2. explain why a current-carrying conductor can have a nonzero internal Electric Field;
3. derive R = ρL∕A and V = IR from local conduction;
4. connect a simple drift model to conductivity;
5. interpret the slope of a current-versus-voltage graph;
6. account for temperature and distinguish ohmic from non-ohmic devices;
7. calculate resistor heating and verify a Power rating;
8. distinguish two-wire and four-wire measurements, and avoid meter-loading errors in EDL01.

1 From the electric field to moving charge

Consider a uniform metal rod connected between two terminals. If the left terminal is at higher electric potential, the electric field inside the rod points approximately from left to right. Electrons have charge −e and drift slowly to the left. Their motion produces conventional current toward the right, as discussed in E02.

The average electron drift is small compared with the electrons’ random microscopic motion. In a metal, interactions with the lattice, impurities, and other scattering processes limit the directed drift. The resulting current can remain steady even though an electric field exerts a force on each carrier.

There is an important distinction from the electrostatics of E01. In an ideal conductor in electrostatic equilibrium, E = 0 inside the bulk. A resistive conductor carrying a steady current is not in electrostatic equilibrium: it generally has E≠0 within the material.

2 Current density and the local form of Ohm’s law

Current density J measures conventional current per unit area and points along the direction of positive charge transport. In a uniform wire of cross-sectional area A, with uniform current density parallel to its axis,

(3)

The full relation from E02 is

(4)

For many isotropic conductors at fixed temperature, over a useful range of fields, experiments find that current density is proportional to the field:

(5)

Here σ is the electrical conductivity. Its SI unit is

(6)

where S denotes the siemens.

The reciprocal is the electrical resistivity ρ:

(7)

Consequently,

(8)

Resistivity has SI unit Ω m. A material with low resistivity can carry more current at the same electric field than a material with high resistivity. For a homogeneous isotropic material, ρ is a scalar; more complex materials may require conductivity and resistivity tensors.

3 Deriving resistance from material and geometry

Take a straight, uniform conducting specimen of length L and cross-sectional area A. Assume that ρ is constant within the specimen, the current density is approximately uniform, and the DC conduction field is axial. From the potential-field relation of E01,

(9)

With a uniform field directed along the specimen,

(10)

Similarly, the current is

(11)

Substitute E = ρJ:

V	= EL,	(12)
	= ρJL,	(13)
	= ρL,	(14)
	= I.	(15)

The quantity multiplying the current is the specimen’s electrical resistance:

(16)

Therefore,

(17)

The resistance is a property of the particular object, including its geometry. Resistivity is a property of the material under specified conditions. The formula R = ρL∕A is not universal for arbitrary shapes, nonuniform fields, contact resistances, or strongly nonlinear conductors.

3.1 Units and a numerical example

Since R = V∕I, its unit is the ohm:

(18)

At approximately 20 ^∘C, a representative resistivity for copper is 1.68 × 10⁻⁸ Ω m. A 10.0 m length of copper wire with diameter 1.00 mm has area

(19)

Its approximate resistance is

R	= ρ,	(20)
	=  Ω,	(21)
	≈ 0.214 Ω .	(22)

The result is a resistance for the specified length, not a universal “resistance of copper.”

4 Why conductivity arises: a simple drift model

A basic microscopic model helps explain the observed proportionality. Let a carrier have mass m_c, signed charge q, and an average momentum relaxation time τ. In the simplest collision model, the mean drift velocity obeys

(23)

In steady state, dv_d∕dt = 0, so

(24)

With carrier number density n, E02 gives

(25)

Combining the expressions,

(26)

Thus this simplified model predicts

(27)

For electrons, q = −e, but q² is positive. An electron drifts opposite to the field while the conventional current density points along the field.

This is a useful model, not a complete theory of metallic conduction. Real materials require additional microscopic physics to predict the actual value and temperature dependence of σ.

5 What Ohm’s law says, and what it does not say

If R is constant over an experimental range, then I = V∕R. A graph of current I on the vertical axis against voltage V on the horizontal axis is a straight line through the origin with slope

(28)

By contrast, if voltage is plotted vertically against current, the slope is R. One must read the axis labels before interpreting a measured slope.

Ohmic behavior means an approximately linear current-voltage relationship at fixed physical conditions. It is an empirical material or device behavior, not a universal consequence of conservation of charge. A diode, heated lamp filament, and many semiconductor devices are non-ohmic under ordinary operating conditions.

For nonlinear devices, two different quantities are often useful:

(29)

and

(30)

The first measures the ratio at one operating point (when I≠0), while the second describes the local incremental response. These need not be equal. For a linear ohmic resistor, both reduce to the same constant R.

6 How temperature changes resistance

Resistivity depends on temperature. Over a sufficiently narrow range, the resistivity of many metals is approximately

(31)

where ρ₀ is the resistivity at reference temperature T₀ and α is a temperature coefficient in K⁻¹. If changes in specimen dimensions are negligible, the same approximation gives

(32)

For copper near room temperature, a representative value is α ≈ 3.9 × 10⁻³ K⁻¹. With a 100 Ω reference resistance at 20 ^∘C, the model predicts at 60 ^∘C:

R	≈ (100)[1 + (0.0039)(60 − 20)] Ω,	(33)
	≈ 115.6 Ω .	(34)

For a metal, greater thermal scattering commonly reduces carrier mobility. Some semiconductors and thermistors instead exhibit decreasing resistance with increasing temperature. In general, a resistor also warms due to its own electrical power dissipation, so an apparently nonlinear I-versus-V curve may result from changing temperature during a measurement.

7 Conductance and power dissipation

The reciprocal of resistance is called conductance:

(35)

Its SI unit is the siemens, S = Ω⁻¹. Conductance is a property of a component. Conductivity, σ, is a material property with units S∕m.

Electric power delivered to a two-terminal component is P = V I when the current enters the terminal labeled at higher voltage (passive sign convention). For an ohmic resistor,

(36)

This electrical energy is predominantly transferred into thermal energy. E04 develops the power and energy bookkeeping in detail.

For example, applying 5.0 V across a 1.0 kΩ resistor produces

I	= = 5.0 mA,	(37)
P	= = 0.025 W.	(38)

This is below a 0.25 W nominal resistor rating, although appropriate margins and environmental derating still matter.

8 Measuring resistance in the laboratory

8.1 Voltage-current measurement

A direct experimental estimate of resistance is

(39)

A voltmeter measures voltage in parallel with the component; an ammeter measures current in series. A digital multimeter (DMM) typically has a large finite input resistance in voltage mode, and a finite internal burden in current mode. Therefore both instruments can perturb the circuit, particularly at very large or very small resistances.

With the indicated arrangement, the ammeter reads the resistor current plus the small current drawn by the voltmeter. Thus V∕I_meter slightly underestimates the resistor’s actual resistance. For typical DMM inputs and kilohm resistors, the effect may be small; for high resistances, it can matter.

8.2 Ohmmeter mode and unwanted parallel paths

A DMM in ohmmeter mode applies an internal test stimulus and senses the response. Disconnect the power source and discharge capacitors before using resistance mode. Whenever possible, isolate one end of the component because other circuit paths may appear in parallel.

For example, if the intended 1.0 kΩ resistor remains in parallel with another 1.0 kΩ resistor, the in-circuit reading is approximately

(40)

A 500 Ω reading would not imply that either component is defective.

8.3 Two-wire and four-wire measurements

At small resistances, the leads and contacts add a measurable contribution:

(41)

In a four-wire or Kelvin measurement, one pair of conductors carries the test current and a separate high-input-resistance pair senses voltage across the specimen. In the ideal limit,

(42)

High input impedance minimizes current in the voltage-sensing leads. Real four-terminal setups still require attention to contact placement, offset voltages, heating, and instrument specifications.

9 EDL01 bridge: a low-voltage Ohm’s-law experiment

A suitable first experiment uses a 1.0 kΩ, 0.25 W resistor, a current-limited adjustable DC supply (or suitable batteries), and a DMM. Use a low-voltage range such as 0 to 5 V. Verify resistor value with power disconnected, then wire the powered circuit. Do not use mains voltage.

1. Record the resistor’s nominal value, tolerance, and the DMM range.
2. Set the source to 0 V and connect the resistor and meter following the proper series or parallel measurement configuration.
3. Measure the resistor voltage and the associated current at successive settings from 0 to 5 V.
4. Plot I versus measured V _R. Fit the nonzero-current data to I = aV _R + b, checking whether b is consistent with zero.
5. Estimate resistance from R ≈ 1∕a when b is negligible and instrument loading is small.
6. Compare the result with the ohmmeter reading and resistor tolerance.

A reasonable ideal expectation is

(43)

At 5.0 V, this gives 5.0 mA and 25 mW. Do not assume that the supply-panel display equals the measured voltage across the resistor. Wires, source impedance, and meter burden may produce additional voltage drops.

10 Common misconceptions

• A resistor does not consume current. Charge continuity still holds; electrical energy is transferred to other forms.
• R and ρ are not interchangeable. Resistance includes object geometry; resistivity describes material response.
• The equation V = IR is not a universal law for every device at all currents, voltages, and temperatures.
• The slope of an I-versus-V graph is 1∕R, not R.
• A steady current in an ordinary metal generally requires a nonzero internal electric field.
• A DMM changes the circuit to some extent; its voltage, current, and resistance modes are not interchangeable.
• Measuring resistance on a powered circuit is unsafe and can damage the meter.

11 Summary and next steps

The core connections are

(44)

(45)

For simple drift kinetics, σ = nq²τ∕m _c explains why density and mobility matter. Temperature, nonuniform geometry, nonlinear materials, and instrument loading determine how accurately the simplified relations represent a particular experiment.

E03E1 provides worked problems and solutions. E04 will examine electrical power and energy in greater detail; E05-E08 will expand to connected circuits and nonideal sources. EDL01 supplies the related measurements.

References

References