Current Electricity Class 12 Physics Chapter 3 Notes

Electric Current

Electric current is a fundamental concept in the study of electricity. It refers to the flow of electric charge through a conductor. Currents can occur naturally or be controlled in various devices and systems, making them a crucial aspect of our understanding of electrical phenomena.

Here’s a breakdown of key points about electric current:

Direction of Current: Electric current is defined as the rate of flow of electric charge through a conductor. The direction of current is conventionally taken as the direction of the flow of positive charges. In reality, electric current involves the movement of both positive and negative charges, but for simplicity, we use the direction of positive charge flow.
Current Measurement: Current is measured in amperes (A), which is often abbreviated as “amps.” One ampere is defined as one coulomb of charge flowing through a conductor in one second. Mathematically, it is expressed as: I=tq where I is the current in amperes, q is the charge in coulombs, and t is the time in seconds.
Types of Current: There are two main types of electric current:
- Steady Current (Direct Current – DC): In steady current, charges flow continuously in one direction through a conductor. Batteries and most electronic devices operate on DC.
- Alternating Current (AC): In alternating current, the direction of charge flow reverses periodically, typically following a sinusoidal waveform. AC is commonly used for electricity distribution in homes and businesses.
Current Direction: Conventional current flows from the positive terminal to the negative terminal of a voltage source (such as a battery). This convention was established before the discovery of the electron, which is negatively charged. Electrons actually flow in the opposite direction, from the negative to the positive terminal.
Current Density: Current density (J) is a vector quantity that represents the amount of current passing through a unit area. It is useful for describing how current is distributed within a conductor.
Measuring Instruments: Ammeters are devices used to measure electric current in a circuit. They are connected in series with the circuit under test. Instruments like multimeters can measure both current and voltage.
Unit of Current: The SI unit of electric current is the ampere (A). One ampere is equivalent to one coulomb of charge passing through a conductor per second.

Electric Currents in Conductors

Electric charge experiences a force in the presence of an electric field.
Free charges move when subjected to an electric field, contributing to electric current.
In nature, free charged particles exist in the ionosphere, but electrons and nuclei are bound in atoms and molecules.
Bulk matter consists of closely packed molecules, where electrons may or may not be free to move.
Conductors, particularly metals, allow electrons to move freely and develop electric currents when an electric field is applied.

Conductors and Electric Currents
- In solid conductors, negatively charged electrons carry the electric current.
- Electrolytic solutions also conduct electricity but involve both positive and negative charges.
Electron Motion in Conductors
- Without an electric field, electrons move randomly due to thermal motion and collide with fixed ions.
- After collisions, electron speeds remain the same, but their directions become random.
- On average, the number of electrons moving in any direction equals those moving in the opposite direction, resulting in no net electric current.
Impact of an Electric Field
- When an electric field is applied to a conductor, the electrons are accelerated in the direction of the field.
- This acceleration causes the electrons to move and constitutes an electric current.
- The current lasts for a brief period until the charges are neutralized.
Maintaining Steady Electric Fields
- To maintain a steady electric field in a conductor, mechanisms like cells or batteries can supply fresh charges.
- Steady electric fields result in continuous electric currents.

Ohm’s Law and Resistance

About Ohm’s Law
- Discovered by G.S. Ohm in 1828.
- Relates current (I) and potential difference (V) in a conductor.
- Ohm’s Law: V ∝ I or V = R * I
- Resistance (R) is a constant of proportionality measured in ohms (Ω).
Factors Affecting Resistance
- Resistance depends on both the material and dimensions of the conductor.
- Resistance varies with the dimensions:
  - Doubling the length (l) doubles the resistance: R ∝ l
  - Halving the cross-sectional area (A) doubles the resistance: R ∝ 1/A
- Combining length and cross-sectional area: R ∝ l/A
Resistivity (ρ)
- Introduces resistivity (ρ), a material-dependent constant: R = ρ * (l/A).
- Resistivity ρ depends on the material but not on its dimensions.
Ohm’s Law in Terms of Current Density (j)
- Current per unit area (I/A) is called current density (j).
- Ohm’s Law using current density: V = j * ρ * l
Vector Form of Ohm’s Law
- Current density (j) is a vector, directed along the electric field (E).
- Ohm’s Law in vector form: E = j * ρ
- Conductivity (σ), the reciprocal of resistivity: σ = 1/ρ
- Alternate form of Ohm’s Law: j = σ * E

Drift of Electrons and the Origin of Resistivity

Electron Motion and Random Collisions
- Electrons in a conductor experience random collisions with fixed ions.
- After collisions, electrons emerge with the same speed but random directions.
- The average velocity of electrons is zero: ∑ vi/N = 0.
Effect of an Electric Field
- In the presence of an electric field (E), electrons are accelerated.
- Electron acceleration due to E: a = -eE/m.
Average Velocity and Drift Velocity
- Consider an electron that experienced its last collision at time ti.
- Velocity at time t: Vi = vi + (-eE/m) * ti.
- Average velocity of electrons at time t is the average of all Vi’s.
- Average velocity (vd) is constant with time: vd = -eEt/m.
Drift Velocity and Net Transport of Charges
- Drift velocity (vd) is the constant average velocity of electrons due to the electric field.
- Electrons with vd move across an area perpendicular to E, causing a net transport of charges.
- Charge transported across area A in time Dt is -neA|vd|Dt.
- Current (I) is the magnitude of the charge transported across the area in time Dt.
Relation Between Current Density and Drift Velocity
- Current density (j) is related to I by I = |j|A.
- jE = τne|vd|m.
Ohm’s Law and Conductivity
- jE = σE, with σ = τne/m.
- Conductivity (σ) is related to relaxation time (τ), electron density (n), and electron mass (m).
Limitations of Ohm’s Law
- Assumptions: τ and n are constants, independent of E.
- In reality, the behavior may deviate from Ohm’s Law under certain conditions.

Mobility

Conductivity depends on the mobility of charge carriers.
Different materials have mobile charge carriers like electrons, ions, or both.
Mobility (µ) is a critical parameter in understanding the flow of charge carriers.

Definition of Mobility
- Mobility (µ) is defined as the magnitude of the drift velocity (vd) per unit electric field (E):
- µ = |vd| / |E|.
- SI unit of mobility is m²/Vs, which is 10⁴ times the practical unit of mobility (cm²/Vs).
- Mobility is always positive.
Mobility Formula
- vd = (eEτ) / m.
- Substituting into the mobility definition:
- µ = |(eEτ) / m|.
- Where τ is the average collision time for electrons.

Limitations of Ohm’s Law

Ohm’s Law, V = RI, is a fundamental relationship for many electrical circuits.
However, Ohm’s Law has limitations and does not hold for all materials and devices.

Types of Deviations from Ohm’s Law
- (a) Non-Proportional V and I:
  - In some materials or devices, V is not directly proportional to I.
- (b) Dependence on the Sign of V:
  - The relationship between V and I depends on the sign of V.
  - Reversing the direction of V while keeping its magnitude fixed does not produce the same current magnitude in the opposite direction.
  - This behavior is observed in diodes.
- (c) Non-Unique Relationship:
  - Some materials exhibit multiple values of V for the same current I.
  - An example is GaAs (gallium arsenide).
Usage of Non-Ohmic Materials and Devices
- Materials and devices that do not follow Ohm’s Law are commonly used in electronic circuits.
- Understanding their behavior is crucial for specific applications.
Focus on Ohmic Materials
- In the upcoming chapters, the focus will be on materials that obey Ohm’s Law.
- These materials have a linear relationship between voltage and current, making analysis more straightforward.

Resistivity of Various Materials

Classification of Materials
- Materials are classified into three main categories based on their resistivities, in increasing order:
  - Conductors
  - Semiconductors
  - Insulators
Conductors
- Conductors have low resistivities, typically ranging from 10⁻⁸ Ωm to 10⁻⁶ Ωm.
- Examples of conductors include most metals.
Insulators
- Insulators have very high resistivities, often exceeding 10¹⁸ times that of metals.
- Common insulators include ceramics, rubber, and plastics.
Semiconductors
- Semiconductors fall between conductors and insulators in terms of resistivity.
- A characteristic of semiconductors is that their resistivity tends to decrease as temperature increases.
- The resistivities of semiconductors can also be intentionally altered by adding small amounts of specific impurities.
- This property is crucial for utilizing semiconductors in electronic devices, as it allows precise control of their electrical behavior.

Temperature Dependence of Resistivity

The resistivity of a material exhibits temperature dependence.
Different materials have varying degrees of temperature dependence.

Temperature Coefficient of Resistivity
- Over a limited temperature range, the resistivity of a metallic conductor is approximately given by:
  - ρT = ρ₀ [1 + a (T – T₀)]
- ρT is the resistivity at temperature T, ρ₀ is the resistivity at reference temperature T₀, and a is the temperature coefficient of resistivity.
- The dimension of a is (Temperature)^(-1), and for metals, a is typically positive.
- A graph of ρT vs. T is approximately a straight line within a limited temperature range.
Materials with Weak Temperature Dependence
- Some materials like Nichrome, Manganin, and Constantan exhibit weak dependence of resistivity on temperature.
- These materials are suitable for wire-bound standard resistors because their resistance changes very little with temperature.
Temperature Dependence in Semiconductors
- Unlike metals, semiconductors have a decrease in resistivity with increasing temperature.
- The resistivities of semiconductors decrease as temperature rises.
- This temperature dependence is typically pronounced in semiconductors.
Understanding Temperature Dependence
- The temperature dependence of resistivity can be understood qualitatively based on the equation ρ = 1 / (n e μ).
- Resistivity ρ depends inversely on the number of free electrons per unit volume (n) and the average time between collisions (τ).
- As temperature increases, the average speed of electrons increases, leading to more frequent collisions and a decrease in τ.
- In metals, n remains relatively constant, so the decrease in τ causes an increase in resistivity.
- In insulators and semiconductors, n increases with temperature, which more than compensates for the decrease in τ, resulting in a decrease in resistivity.

Electrical Energy and Power

Consider a conductor with a current (I) flowing from point A to point B, where the electric potential at A and B is denoted as V(A) and V(B), respectively.
Electric potential difference (V) exists between A and B due to the current flow.

Potential Energy Change
- In a time interval (Δt), a charge (ΔQ = IΔt) travels from A to B.
- The potential energy of the charge at A is QV(A), and at B, it is QV(B).
- The change in potential energy (ΔUpot) is given by ΔUpot = ΔQ(V(B) – V(A)) = -ΔQV = -IVΔt.
- Charges moving without collisions would change their kinetic energy, but collisions with ions and atoms lead to energy dissipation as heat.
Conservation of Energy
- In an ideal conductor without collisions, the kinetic energy of charges would increase as they move (ΔK = IVΔt).
- However, in actual conductors with collisions, this energy is dissipated as heat (ΔW = IVΔt).
Power Dissipation
- The power dissipated as heat is P = ΔW/Δt = IV.
- Using Ohm’s law (V = IR), P can be expressed as P = I²R or P = V²/R .
- This power loss, known as “ohmic loss,” is responsible for heating elements such as the coil in an electric bulb.
Source of Power
- To maintain a steady current through a conductor, an external source (e.g., a battery or generator) supplies the power.
- In the circuit, chemical energy from the cell supplies the required power.
Power Transmission
- In power transmission, minimizing power loss in transmission cables is essential.
- Transmission cables have a resistance (Rc).
- The power wasted in the connecting wires (Pc) is proportional to I²Rc and inversely proportional to V².
- To reduce Pc, high-voltage transmission lines are used, which necessitates transformers to lower the voltage for safe usage.

Cells in Series and Parallel

Series Combination of Cells:

Cells can be combined in electric circuits similar to resistors.
Like resistors, cells in series can be replaced by an equivalent cell.
Consider two cells in series:
- e1, e2: EMFs of the cells
- r1, r2: Internal resistances of the cells
Potentials at points A, B, and C: V(A), V(B), V(C)
Potential difference between A and B: V(A) – V(B) = ε1
Potential difference between B and C: V(B) – V(C) = ε2
Potential difference between A and C: V(A) – V(C) = ε1 – ε2 – Ir
To replace the combination with a single cell (A to C):
- V(AC) = eeq – Ireq
Equivalent EMF (eeq) for series combination: eeq = e1 + e2
Equivalent internal resistance (req) for series combination: req = r1 + r2
If the current leaves any cell from the negative electrode, use a negative sign in the equation for eeq.

Parallel Combination of Cells:

Consider a parallel combination of cells:
- I1, I2: Currents leaving the positive electrodes
- I: Total current flowing out
Potential difference across the first cell: V(B1) – V(B2) = ε
Similar potential difference for the second cell: V(B1) – V(B2) = ε
Total current: I = I1 + I2
Potential difference across the combination: V = ε1 – ε2 = I(r1 + r2)
To replace the combination with a single cell (B1 to B2):
- V = eeq – Ireq
Equivalent EMF (eeq) for parallel combination: 1/eeq = 1/e1 + 1/e2 + …
Equivalent internal resistance (req) for parallel combination: 1/req = 1/r1 + 1/r2 + …

General Equations for n Cells in Parallel:

For n cells with emf e1, e2, …, en and internal resistances r1, r2, …, rn:
Equivalent EMF (eeq): 1/eeq = 1/e1 + 1/e2 + … + 1/en
Equivalent internal resistance (req): 1/req = 1/r1 + 1/r2 + … + 1/rn

Wheatstone Bridge and Its Balance Condition

The Wheatstone bridge is an electrical circuit used for measuring unknown resistances.
It consists of four resistors: R1, R2, R3, and R4.
The bridge is powered by a source connected across diagonally opposite points (A and C) known as the battery arm.
A galvanometer (G) is connected between the other two vertices (B and D) called the galvanometer arm.

Balanced Bridge Condition:

In the balanced bridge condition, the galvanometer shows zero current (Ig = 0), indicating that there is no current through G.
To achieve balance, we start by applying Kirchhoff’s junction rule at junctions D and B, yielding the relations I1 = I3 and I2 = I4.
Next, we apply Kirchhoff’s loop rule to closed loops ADBA and CBDC.

Loop Equations:

Loop ADBA:
- -I1R1 + 0 + I2R2 = 0 (Ig = 0)
- I1R1 = I2R2
Loop CBDC:
- I2R4 + 0 – I1R3 = 0 (I3 = I1, I4 = I2)
- I2R4 = I1R3

Balance Condition:

Equ 1: I1/I2 = R2/R1
Equ 2: I2/I1 = R3/R4
Combining the two equations:
- (R1/R2) * (R4/R3) = 1

Balance Condition for the Wheatstone Bridge:

2/1 * 4/3 = R/R4 = R1/R2 * R4/R3 = 1

Practical Application: Determination of Unknown Resistance R4:

The Wheatstone bridge provides a practical method for measuring an unknown resistance (R4).
Assume R4 is unknown, while R1 and R2 are known resistances.
Vary R3 until the galvanometer shows null deflection, indicating a balanced bridge.
From the balance condition, the value of the unknown resistance R4 is given by:
- R4 = (R2/R1) * R3
A practical device based on this principle is called the meter bridge, used for precise resistance measurements.