Ohm’s law

What is Ohm’s law?

In plain words: push harder (more voltage) and more current flows, in exact proportion. The resistance says how hard the wire resists: a longer wire resists more, a thicker one less, and the material sets the rest through its resistivity.

V = RI

• Ohm’s law was found by G. S. Ohm long before the mechanism of current flow was understood (NCERT page 83). NCERT dates it to 1828; Ohm did the work in 1825–26 and set the law out in full in his book Die galvanische Kette, mathematisch bearbeitet (Berlin, 1827; a first edition is digitised on the Bibliothèque nationale de France’s Gallica library).
• For a conductor with current I and potential difference V across its ends (NCERT Eq. (3.3), page 83):

• R is the resistance, measured in ohms; 1 Ω = 1 V A⁻¹ (from R = V/I).
• V = RI on its own only defines R = V/I, for a nonzero current; it is then a static ratio even for a non-ohmic device. Ohm’s law is the statement that R does not depend on V, so that the V–I graph is a straight line through the origin; in the form E = ρj, a material obeys it when ρ does not depend on the size or direction of the field (NCERT Points to Ponder 2, page 105).
• Ohm was led to his law by an analogy with the conduction of heat: the electric field plays the part of the temperature gradient, and the current the part of the heat flow (NCERT’s box on Ohm, page 84; see heat conduction). (The box gives his birth year as 1787; standard references, such as Encyclopaedia Britannica, give 16 March 1789.)

Resistance and resistivity

The formula R = ρl/A assumes a uniform wire of constant cross-section and uniform scalar resistivity. The scalar forms E = ρj and j = σE apply to an isotropic material; I/A is the local current density only when the flow is uniform across the section (otherwise it is the cross-sectional average).

NCERT finds how R depends on the size of a conductor with a slab of length l and cross-section A (NCERT pages 83–84, Figure 3.2):

• Two slabs end to end (length 2l): the same current I flows through both, and the potential differences add to 2V. So the combination has R_C = 2V/I = 2R (NCERT Eq. (3.4)). Doubling the length doubles the resistance: R ∝ l (Eq. (3.5), page 84).
• The slab cut lengthwise into two of area A/2: with the same V, each half carries I/2, so each has R₁ = V/(I/2) = 2R (NCERT Eq. (3.6)). Halving the area doubles the resistance: R ∝ 1/A (Eq. (3.7), page 84).
• Combining the two, R ∝ l/A (NCERT Eq. (3.8)), so (NCERT Eq. (3.9), page 84):

• The constant ρ is the resistivity. It depends on the material and its temperature, pressure and other physical conditions, but not on the dimensions (page 84; see resistivity of various materials). Its unit is Ω m (from ρ = RA/l). Ohm’s law then reads V = IR = Iρl/A (NCERT Eq. (3.10)).

Current density

NCERT defines it:

Current per unit area (taken normal to the current), I/A, is called current density and is denoted by j.

NCERT Physics Part I (Class 12), Section 3.4, page 84

• Its SI unit is A/m² (page 84).
• In a uniform field E along a conductor of length l, V = El. Putting this and I = jA into V = Iρl/A gives El = jρl, so E = jρ (NCERT Eq. (3.11), page 85).
• j is a vector along E, so in vector form E = ρj (NCERT Eq. (3.12), page 85).

Conductivity

• The conductivity is σ = 1/ρ, and Ohm’s law can be written (NCERT Eq. (3.13), page 85):

• This form, j = σE, is often used as an equivalent statement of Ohm’s law, alongside V = RI (page 85). It works at each point inside a conductor, not just between its ends (reasoned from NCERT Eq. (3.12)).
• The unit of σ is Ω⁻¹ m⁻¹ (from σ = 1/ρ).

Solved examples

Common mistakes to avoid

Quick revision summary

• Ohm’s law: V = RI, with R the resistance in ohms; R depends on the material and on the dimensions.
• Doubling the length doubles R; halving the area doubles R; so R = ρl/A, with ρ the resistivity of the material, in Ω m.
• Current density j = I/A, in A/m^², is a vector along E; E = ρj.
• Conductivity σ = 1/ρ, and Ohm’s law can be written j = σE.
• A wire stretched to n times its length, at fixed volume, has n^² times the resistance (R ∝ 1/r^⁴); at unchanged resistivity a 1 % stretch raises R by about 2 %.

Questions students ask

What is the difference between resistance and resistivity?

Resistance R belongs to a particular conductor and is measured in ohms, while resistivity ρ, also called specific resistance, belongs to the material and is measured in Ω m: R = ρl/A grows with the conductor’s length and falls with its cross-section, but ρ depends only on the material and on its temperature and pressure (NCERT Eq. (3.9), page 84; NCERT Summary point 4, page 103). Numerically, ρ is the resistance between opposite faces of a cube of the material 1 m on a side, since for that cube l/A = 1 m^⁻¹.

Why is the unit of resistivity Ω m and not Ω per metre?

Resistivity is measured in ohm metres because ρ = RA/l multiplies a resistance by an area and divides by a length, Ω × m^²/m = Ω m (from NCERT Eq. (3.9), page 84); its dimensional formula, [M L^³ T^⁻³ A^⁻²], has one more power of L than that of resistance, [M L^² T^⁻³ A^⁻²] (NCERT table, page 104; see dimensional formulae). Ω m^⁻¹ is the unit of a different quantity, the resistance per unit length of a particular wire, R/l = ρ/A, which still depends on how thick the wire is. A wire of resistivity 1.0 × 10^⁻⁶ Ω m has 1.0 Ω m^⁻¹ if its cross-section is 1.0 mm^², but 4.0 Ω m^⁻¹ if it is 0.25 mm^².

Is conductance the same as conductivity?

No, conductance and conductivity are different quantities: conductance G = 1/R belongs to a particular conductor and is measured in siemens, 1 S = 1 Ω^⁻¹, while conductivity σ = 1/ρ belongs to the material and is measured in S m^⁻¹, or Ω^⁻¹ m^⁻¹ (NCERT Eq. (3.13), page 85; NCERT Chemistry Part I (Class 12), Unit 2, page 41). They are linked as resistance and resistivity are, G = σA/l. The table on page 104 of NCERT Physics Part I prints the unit of conductivity as S, the unit of conductance, though the dimensions it gives, [M^⁻¹ L^⁻³ T^³ A^²], are those of S m^⁻¹.

Is Ohm’s law derived from theory or found by experiment?

Ohm’s law was found by experiment: G. S. Ohm discovered it, in 1828 according to NCERT, long before anyone knew how charges move through a metal (NCERT page 83). NCERT later shows that a simple model of electrons drifting between collisions reproduces it, giving j = (ne^²τ/m)E (NCERT Eq. (3.22), page 86), but only by assuming that n and τ do not depend on E (page 87). So the ‘derivation’ explains why many metals obey the law, and where its assumptions fail, so does the law.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A 5.0 Ω resistor carries 2.0 A. What is the potential difference across it?

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   V = RI = 5.0 × 2.0 = 10 V (NCERT Eq. (3.3), page 83).

2. Q2. Two wires of the same material have the same length, but one has twice the diameter of the other. Compare their resistances.

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   Twice the diameter means four times the area, so the thicker wire has one quarter the resistance (NCERT Eq. (3.9), page 84).

3. Q3. What is the SI unit of current density?

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   A/m² (NCERT page 84).