Revision: Current Electricity >> Electric Resistance and Ohm's Law Physics (Theory) ISC (Science) ISC Class 12 CISCE

The resistivity of a material is the resistance of a wire of that material of unit length and unit area of cross-section.

Current is defined as the rate of flow of charge.

A continuous and closed path of an electric current is called an electric circuit.

An electric current is measured by the amount of electric charge moving per unit time at any point in the circuit.

The magnitude of an electric current is the number of electric charges flowing through a conductor in one second.

Electromotive force: When no current is drawn from a cell, when the cell is in open circuit, the potential difference between the terminals of the cell is called its electromotive force (or e.m.f.).

The movement of the positive charge is called conventional current.

The unit of electric current is ampere (A). When 1 C of charge flows through a conductor in 1 s, it called 1 ampere (A) current.
I = `Q/t`

Substances whose resistance decreases tremendously with decreasing temperature and reaches nearly zero near absolute zero are called superconductors; e.g., lead, tin, etc.

 Semiconductors: Substances whose resistance decreases with the increase in temperature are named as semiconductors. E.g. manganin, constantan etc.

Resistance is the obstacle that the wire presents to the current flow.

The obstruction offered to the flow of current by the conductor (or wire) is called its resistance.

A variable resistor has a resistance that can be varied. It is used to vary the amount of current flowing in a circuit.

A fixed resistor has a resistance of a fixed value. Common types of fixed resistors include carbon film resistors and wire-wound resistors.

One coulomb is the amount of electric charge transferred by a current of one ampere in one second.

One ohm is the resistance of a component when the potential difference of one volt applied across the component drives a current of one ampere through it.

The temperature coefficient is defined as the ratio of the increase in resistivity per degree rise in temperature to its resistivity at T₀.

Current density is a vector quantity, often known as an area vector or cross-sectional area vector, whose value is equal to the electric current flowing per unit area.

J = `"I"/"A"`

S.I unit is A/m².

The reciprocal of resistance is called conductllnce. It is denoted by the letter G. 

The conductors which do not obey Ohm's law are called non ohmic resistors (or non-linear resistances).

The conductors which obey Ohm's law are called ohmic resistors (or linear resistances).

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]

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.

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

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 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
  

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.

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]

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].

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.

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 = \[\frac {watt × hour}{1000}\]

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

According to Ohm’s law, the current flowing in a conductor is directly proportional to the potential difference across its ends, provided the physical conditions and temperature of the conductor remain constant.
No, it is not always true. E.g., Diode valve, junction diode, etc., do not obey Ohm’s law.

Statement: Ohm’s Law

"The electric current flowing through a conductor is directly proportional to the potential difference across its ends, provided the temperature and other physical conditions of the conductor remain constant."

Mathematically,

I ∝ V or V = I R

where:

• V = Potential difference (in volts)
• I = Current (in amperes)
• R = Resistance of the conductor (in ohms, Ω)

Explanation:

When two conductors at different electric potentials are joined by a metallic wire, electrons flow from the conductor at a lower potential (excess electrons) to the one at a higher potential (deficit of electrons). This movement of electrons results in an electric current.

• The current continues to flow until both conductors reach the same potential.
• For continuous current flow, a constant potential difference must be maintained across the ends of the conductor (e.g., using a battery or power supply).

Derivation / Mathematical Proof:

From Ohm’s Law:

I ∝ V ⇒ \[\frac {V}{I}\] = constant

This constant is defined as the resistance (R) of the conductor. Therefore,

V = I R   ---(1)

This is the mathematical form of Ohm’s Law.

Special Case:

If the current I = 1 A, then:

V = R

This implies that the resistance of a conductor is numerically equal to the potential difference across it when 1 ampere of current flows through it.

Conclusion:

Ohm's Law provides a fundamental relationship between voltage, current, and resistance in an electric circuit. It is widely used in the design and analysis of electrical and electronic systems.

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.

• Electricity is a convenient and controllable form of energy widely used in homes, industries, schools, and hospitals.
• Electric current is produced when electric charges flow through a conductor, and it flows only through a closed, continuous electric circuit.
• A switch completes or breaks the circuit; when the circuit is broken, current stops flowing, and devices like bulbs do not glow.
• Electric current is the rate of flow of charge, given by the relation I = Q / t, where Q is charge and t is time.
• In metallic wires, electrons are the charge carriers, but by convention, current flows from the positive to the negative terminal, in the opposite direction to electron flow.

• Free electrons in a metal move randomly; without a potential difference, there is no net flow of current.
• When a potential difference is applied, electrons drift towards the positive terminal, but collide with fixed positive ions, losing energy.
• These collisions cause resistance, and the number of collisions determines the amount of resistance in the conductor.

• 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.

• 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.

• 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