Overview: Electric Resistance and Ohm's Law

The charge flowing per second in an electric circuit is the measure of electric current in that circuit.

Mathematically,

I = \[\frac {Q}{t}\]

• 1 ampere = 1 coulomb/second ⇒ 1A = 1Cs^-1
• 1 ampere = 6.25 x 10¹⁸ electrons per second
  

Current density is defined as the current flowing through unit cross-sectional area drawn through that point perpendicular to the direction of flow of current.

Mathematically,
j = \[\frac {I}{A}\]

SI unit = ampere/metre² (A m^-2), Dimensions = [A L^-2].

The average distance moved by a free electron between two successive collisions is called 'mean free path' of the electron.

The average time-interval between two successive collisions is called the 'relaxation time' of the electron.

Drift velocity defined as the average velocity with which the free electrons get drifted towards the positive end of the conductor under the influence of external applied electric field.

OR

The average velocity with which free electrons drift opposite to the direction of the applied electric field.

The mobility of a free electron is numerically equal to the magnitude of drift velocity imparted by a uniform electric field of strength 1 V-m^-1.

SI unit: m²v^-1s^-1.

OR

The mobility m defined as the magnitude of the drift velocity per unit electric field:

μ = \[\frac {v_d}{E}\] = \[\frac {eτ}{m}\]

The ratio of the potential difference to the current is called the ‘electric resistance’ R of the conductor.

Mathematically.
R = \[\frac {V}{I}\]

1 ohm = 1 volt/ampere ⇒ 1Ω = 1VA^-1
Dimensions =  [M L² T^-3A^-2]

If a small change ΔV in the potential difference across a part of a non-ohmic circuit causes a change ΔI in electric current, then the ratio ΔV/ΔI is called the 'dynamic resistance' of that part of the circuit.

Mathematically.
\[\frac {ΔV}{ΔI}\]

The ratio of the intensity of the electric field E at any point within the conductor and the current-density j at that point is called ‘specific resistance' or ‘electrical resistivity' of the conductor and is represented by ρ.

Mathematically,
ρ = \[\frac {E}{j}\]

Dimensions = [M L³ T^-3 A^-2]

The reciprocal of specific resistance is called 'specific conductance' and is represented by σ.

σ = \[\frac {1}{ρ}\]

SI unit = (ohm-metre)^-1 ⇒ (Ω-m)^-1
Dimension = [M^-1 L^-3 T³ A²]

When two or more resistances connected between two points are replaced by a single resistance such that there is no change in the current of the circuit and the potential difference between those two points, the single resistance is called the equivalent resistance.

The potential difference between two points in an electric circuit is defined as the work done in carrying a unit charge from one point to the other.

The rate at which electric energy is transferred into other forms of energy is called ‘electric power’ P.

1 kilowatt-hour, or 1 unit, is the quantity of electric-energy which is dissipated in 1 hour in a circuit when the electric power in the circuit is 1 kilowatt.

P = \[\frac {W}{t}|] = V I

Power in a Resistor:
P = I²R and P = \[\frac {V^2}{R}\]

1 kW-h = 3.6 x 10⁶ W-s = 3.6 × 10⁶ J

Units = \[\frac {watt × hour}{1000}\]

Statement

The variation of current with voltage is the macroscopic form of Ohm’s law. When the situation is considered at a point, the law is known as Ohm’s law in microscopic (vector) form.

Explanation/Proof

From, V = \[\frac{m}{ne^2\tau}\frac{l}{A}I\]

or

\[\frac{V}{l}=\left(\frac{m}{ne^{2}\tau}\right)\left(\frac{I}{A}\right)\]

But,

\[\frac {V}{l}\] = E, \[\frac {m}{n e^2 τ}\] = ρ and \[\frac {I}{A}\] = j,

\[\therefore\] E = ρ j

Also, ρ = \[\frac {1}{σ}\]​

Hence,

E = \[\frac {1}{σ}\]j or j = σ E

In vector notation,

\[\vec j\] = σ\[\vec E\]

Conclusion

Therefore, for an isotropic substance,

\[\vec j\] ∝ \[\vec E\]

and Ohm’s law in vector form states that the current density is directly proportional to the applied electric field strength, and the ratio of current density to electric field is a constant σ, independent of the electric field producing the current.

• Ohm’s law does not hold when temperature changes due to current flow, causing resistance to vary (e.g., filament bulb).
• In some materials, current starts flowing only after a minimum applied voltage, so the V–I graph is not linear.
• Devices like diodes, thermistors, and vacuum tubes are non-ohmic because their resistance is not constant

• Metals: Resistivity increases with a rise in temperature due to increased electron collisions.
• Temperature coefficient: For metals, resistance varies with temperature as
  R_t = R₀(1 + αt),
  and for most metals, α ≈ \[\frac {1}{273}\] per °C, so R ∝ T (approximately).
• Alloys: The resistivity of alloys changes very little with temperature and remains relatively high.
• Semiconductors: Resistivity decreases with an increase in temperature due to an increase in charge carriers.
• Electrolytes: Resistivity decreases with a rise in temperature because ions move more freely.

• Carbon resistors use colour codes to indicate resistance value; the first two bands give significant figures and the third band gives the multiplying power of 10.
• The fourth colour band indicates the resistor tolerance: gold (±5%), silver (±10%), and no band (±20%).
• The colour sequence Black to White represents digits 0 to 9, and the same colours in the third band represent multipliers 10⁰ to 10⁹.

• Series combination: Same current flows through all resistances, and the equivalent resistance is
  R = R₁ + R₂ + R₃​
• Series property: In a series, the equivalent resistance is greater than the largest individual resistance, and the voltage divides in the ratio of resistances.
• Parallel combination: Same potential difference exists across all resistances and the equivalent resistance satisfies
  ​\[\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\]
• Parallel property: In parallel, the equivalent resistance is less than the smallest individual resistance, and current divides inversely with resistance.
• Practical use: Household electrical appliances are connected in parallel, so each works independently at the same voltage.

• Series combination: The net power consumed decreases; for identical bulbs,
  P_consumed = \[\frac {P}{n}\]​
  and it is directly proportional to bulb resistance and inversely proportional to rated power.
• Parallel combination: The net power consumed increases; for identical bulbs,
  P_consumed = n P
  and it is inversely proportional to bulb resistance and directly proportional to rated power.

v_d = \[-\frac{e\mathbf{E}}{m}\tau\]

• Ohm’s law is not valid for all materials; in some devices, voltage is not proportional to current.
• In certain materials (like diodes), reversing the voltage does not produce equal current in the opposite direction.
• Some materials show non-unique V–I characteristics, meaning more than one voltage value may correspond to the same current.