Current Electricity Class 12 – Complete Guide with Concepts, Formulas, NCERT Solutions & CBSE Previous Year Questions

Chapter 3:Current Electricity

3.1 Introduction to Current Electricity

3.2 Electric Current

3.3 Electric Current in Conductors

3.4 Ohm’s Law And Limitations of Ohm’s Law

3.5 Drift of Electrons and the Origin of Resistivity

3.6 Resistivity of Various Materials

3.7 Temperature Dependance of Resistivity

3.8 Electrical Energy, Power

3.9 Combination of Resistors-Series and Parallel

3.10 Cells, emf, Internal Resistance

3.11 Cells in Series and in Parallel

3.12 Kirchhoff’s Rules

3.13 Wheatstone Bridge

3.14 Metre Bridge

3.15 Potentiometer

1. Introduction to Electric Current

• Charges in motion constitute an electric current.
• Naturally occurring currents include phenomena like lightning, where charges flow from clouds to the earth, though this flow is not steady.
• Many everyday devices, such as a torch or a cell-driven clock, involve charges flowing in a steady manner, similar to water flowing smoothly in a river.
• This chapter focuses on the basic laws concerning steady electric currents.

2. Definition of Electric Current

• To define electric current, imagine a small area held normal to the direction of charge flow.
• Both positive and negative charges can flow across this area, both forward and backward.
• Net charge (q) flowing in the forward direction across the area in a time interval t is defined as q = q+ - q–, where q+ is the net positive charge and q– is the net negative charge flowing in the forward direction.
• For a steady current, this net charge q is proportional to t.
• The current across the area (I) in the forward direction is defined as the quotient: I = q/t. A negative value implies current in the backward direction.
• More generally, for non-steady currents, the current at time t across a cross-section of a conductor is defined as the limit of the ratio of net charge (ΔQ) to time interval (Δt) as Δt tends to zero: I(t) = lim (Δt→0) (ΔQ/Δt).
• In SI units, the unit of current is ampere (A).
• One ampere is defined through magnetic effects of currents.
• Currents in domestic appliances are typically on the order of amperes.
• Lightning can carry currents of tens of thousands of amperes, while currents in human nerves are in microamperes.

3. Electric Currents in Conductors

• An electric charge will move, contributing to a current, if it experiences a force from an applied electric field and is free to move.
• Free charged particles exist in nature, such as in the ionosphere.
• In atoms and molecules, electrons and nuclei are bound and generally not free to move.
• In metals (conductors), some electrons are practically free to move within the bulk material. When an electric field is applied, these materials develop electric currents.
• In solid conductors, current is carried by negatively charged electrons against a background of fixed positive ions.
• Other types of conductors, like electrolytic solutions, can have both positive and negative charges moving. The source primarily focuses on solid conductors.

3.1 Electron Motion in Conductors

• Absence of Electric Field:
  • Electrons undergo thermal motion, colliding with fixed ions.
  • After a collision, an electron emerges with the same speed but a completely random direction.
  • On average, the number of electrons traveling in any direction equals those traveling in the opposite direction, resulting in no net electric current.
  • The average velocity of all electrons is zero in the absence of an electric field.
• Presence of Electric Field:
  • An electric field exerts a force on electrons, causing them to accelerate. The acceleration a = -eE/m, where -e is the electron charge and m is its mass.
  • This acceleration causes electrons to drift in a direction opposite to the electric field.
  • Even though electrons accelerate between collisions, they lose their drift speed upon collision and start accelerating again.
  • Therefore, on average, electrons acquire a steady average drift speed rather than continuous acceleration.

4. Ohm's Law

• Discovered by G.S. Ohm in 1828. Georg Simon Ohm (1787–1854), a German physicist, derived his law through an analogy with heat conduction, where the electric field is analogous to the temperature gradient and electric current to heat flow.
• Statement: For a conductor through which current I flows, and V is the potential difference across its ends, Ohm's law states that V is proportional to I (V ∝ I).
• This proportionality is expressed as V = RI, where R is the resistance of the conductor.
• The SI unit of resistance is ohm (Ω).
• Resistance (R) depends on the material of the conductor and its dimensions.
  • Proportional to length (l): R ∝ l. Doubling the length doubles the resistance.
  • Inversely proportional to cross-sectional area (A): R ∝ 1/A. Halving the area doubles the resistance.
  • Combining these, R = ρ(l/A), where ρ (rho) is the resistivity.
• Resistivity (ρ) depends on the material of the conductor but not on its dimensions.
• Current Density (j): Current per unit area (normal to the current). SI units are A/m².
• Relationship between E, j, and ρ: If E is the magnitude of the uniform electric field and l is the length, then V = El. Substituting into Ohm's law (V = Iρl/A) yields E = jρ.
• In vector form, E = jρ or j = σE.
• Conductivity (σ): Defined as σ = 1/ρ.

5. Drift of Electrons and the Origin of Resistivity

• The average velocity of electrons when an electric field is present is called drift velocity (v_d).
• The acceleration (a) of an electron due to the electric field E is a = -eE/m, where e is the charge and m is the mass of the electron.
• The electron's velocity (V_i) at time t after its last collision (at t_i) is V_i = v_i + (-eE/m)t_i, where v_i is its velocity immediately after the last collision.
• Averaging over all electrons, the average initial velocity v_i is zero because directions are random after collision.
• The average time between successive collisions is denoted by τ (tau), known as the relaxation time.
• The drift velocity (v_d) is thus given by v_d = -eEτ/m. This average velocity is surprisingly independent of time, even though electrons are accelerated.
• Magnitude of Current (I): Due to drift, charge is transported across an area A perpendicular to E. If n is the number of free electrons per unit volume, the total charge transported in time Δt is ΔQ = neA|v_d|Δt.
• Since I = ΔQ/Δt, then I = neA|v_d|.
• Current Density (j): Since I = |j|A, substituting |v_d| from the drift velocity equation yields |j| = ne²Eτ/m.
• In vector form, j = (ne²τ/m)E.
• Comparing this with j = σE, the conductivity (σ) is identified as σ = ne²τ/m.
• Consequently, resistivity (ρ) is ρ = m/(ne²τ). This derivation reproduces Ohm's law based on the characteristics of electron drift.

5.1 Drift Speed Examples

• In a copper wire carrying 1.5 A with a cross-sectional area of 1.0 × 10⁻⁷ m², the drift speed of conduction electrons is estimated to be very small, about 1.1 mm/s.
• This drift speed is much smaller than the thermal speeds of copper atoms (about 2 × 10² m/s) and the speed of propagation of the electric field along the conductor (speed of light, 3.0 × 10⁸ m/s).
• Why current is established instantly: The electric field is established almost instantly (at the speed of light) throughout the circuit, causing a local electron drift everywhere. Current establishment doesn't wait for electrons to travel from one end to the other.
• Why large currents with small drift speed and charge: The number density (n) of free electrons in conductors is enormous, roughly 10²⁹ m⁻³.
• Electron paths: In the absence of an electric field, paths are straight lines between collisions. In the presence of an electric field, paths are generally curved.

5.2 Mobility

• Mobility (μ) is an important quantity defined as the magnitude of the drift velocity per unit electric field: μ = |v_d|/E.
• The SI unit of mobility is m²/Vs.
• From the drift velocity equation, mobility is also expressed as μ = eτ/m.

6. Limitations of Ohm's Law

• Ohm's law is not a fundamental law of nature and is not universally valid.
• Deviations from V ∝ I can occur in materials and devices, broadly categorized as:
  • (a) Non-linear relationship: V ceases to be proportional to I (e.g., resistivity increases with current).
  • (b) Dependence on sign of V: The relationship between V and I depends on the sign of V. Reversing the voltage direction does not produce a current of the same magnitude in the opposite direction (e.g., in a diode).
  • (c) Non-unique relationship: There is more than one value of V for the same current I (e.g., GaAs).
• Devices not obeying Ohm's law are widely used in electronic circuits.

7. Resistivity of Various Materials

• Materials are classified into conductors, semiconductors, and insulators based on their resistivities in increasing order.
• Conductors (Metals): Have low resistivities, typically in the range of 10⁻⁸ Ωm to 10⁻⁶ Ωm. Examples include copper (Fig. 3.8 shows its resistivity vs. temperature).
• Insulators: Have very high resistivities, 10¹⁸ times greater than metals or more. Examples include ceramic, rubber, and plastics.
• Semiconductors: Fall between conductors and insulators.
  • Their resistivities characteristically decrease with a rise in temperature (Fig. 3.10).
  • Their resistivities can be decreased by adding small amounts of suitable impurities, a feature exploited in electronic devices.

8. Temperature Dependence of Resistivity

• The resistivity of a material depends on temperature, and different materials show different dependencies.
• For a metallic conductor over a limited temperature range, resistivity is approximately given by: ρ_T = ρ_0[1 + α(T - T_0)].
  • ρ_T is resistivity at temperature T, and ρ_0 is resistivity at reference temperature T_0.
  • α (alpha) is the temperature coefficient of resistivity, and its dimension is (Temperature)⁻¹.
  • For metals, α is positive, meaning resistivity increases with temperature.
  • The graph of ρ_T vs. T deviates from a straight line at temperatures much lower than 0°C.
• Explanation for metals: As temperature increases, the average speed of electrons increases, leading to more frequent collisions with fixed ions. This causes the average time between collisions (τ) to decrease. Since the number of free electrons (n) in a metal is not appreciably dependent on temperature, the decrease in τ causes ρ to increase.
• For insulators and semiconductors:
  • Resistivities decrease with increasing temperatures.
  • This is because the number of free electrons (n) increases with temperature. This increase in n more than compensates for any decrease in τ, leading to a decrease in resistivity.
• Alloys (e.g., Nichrome, Manganin, Constantan): Exhibit a very weak dependence of resistivity with temperature (Fig. 3.9). These materials are used in wire-bound standard resistors because their resistance values change very little with temperature.

9. Electrical Energy and Power

• When current I flows from point A to B in a conductor, where V(A) > V(B), the potential energy of the charge decreases.
• This lost potential energy is dissipated as heat in the conductor due to collisions of charge carriers with atoms and ions. The atoms vibrate more vigorously, causing the conductor to heat up.
• The energy dissipated as heat (ΔW) in time interval Δt is: ΔW = IVΔt.
• The power dissipated (P), which is energy dissipated per unit time, is: P = IV.
• Using Ohm's law (V = IR), power can also be expressed as P = I²R or P = V²/R. This is known as ohmic loss.
• This power heats up components, like the coil of an electric bulb to incandescence, radiating heat and light.
• The power for maintaining a steady current comes from an external source, such as the chemical energy of a cell.

9.1 Power Transmission

• Minimizing power loss in transmission cables is crucial for transmitting electrical power from power stations to homes and factories.
• Power wasted in connecting wires (P_c) with resistance R_c is P_c = I²R_c = (P/V)²R_c.
• To reduce P_c, transmission cables carry current at enormous values of voltage (V), because power wasted is inversely proportional to V².
• This is why high voltage danger signs are common on transmission lines. Transformers are used at the other end to lower the voltage to a safe value for use.

10. Cells, EMF, Internal Resistance

• An electrolytic cell maintains a steady current in a circuit.
• A cell has two electrodes (positive 'P' and negative 'N') immersed in an electrolyte.
• Electromotive Force (emf, ε): This is the potential difference between the positive and negative electrodes in an open circuit (i.e., when no current is flowing through the cell).
  • It is calculated as the sum of potential differences at each electrode-electrolyte interface: ε = V+ + V–.
  • Note: Emf is a potential difference, not a force, despite its historical name. It is the work done per unit charge by the source to move charge from lower to higher potential energy.
• Internal Resistance (r): The electrolyte through which current flows has a finite resistance 'r', called the internal resistance.
• Terminal Voltage (V) when current flows: When an external resistance R is connected and current I flows, the potential difference between the terminals of the cell is V = ε - Ir.
  • The term Ir represents the voltage drop across the internal resistance.
• Combining with Ohm's Law (V = IR) for the external circuit, the current flowing is I = ε / (R + r).
• The maximum current that can be drawn from a cell is I_max = ε/r (when R = 0), though this maximum is often limited to prevent cell damage.
• Internal resistances of dry cells are generally much higher than common electrolytic cells.

11. Cells in Series and in Parallel

• Combinations of cells can be replaced by an equivalent cell for circuit calculations.

11.1 Cells in Series

• When n cells are connected in series (positive of one to negative of the next, and so on):
  • Equivalent emf (ε_eq): The sum of individual emfs: ε_eq = ε₁ + ε₂ + ... + ε_n.
  • Equivalent internal resistance (r_eq): The sum of individual internal resistances: r_eq = r₁ + r₂ + ... + r_n.
• If current leaves any cell from its negative electrode in the combination, that cell's emf enters the equivalent emf expression with a negative sign.

11.2 Cells in Parallel

• When n cells are connected in parallel (all positive terminals together, and all negative terminals together):
  • Equivalent internal resistance (r_eq): 1/r_eq = 1/r₁ + 1/r₂ + ... + 1/r_n.
  • Equivalent emf (ε_eq): ε_eq/r_eq = ε₁/r₁ + ε₂/r₂ + ... + ε_n/r_n.
  • These equations are valid even if a cell is connected in reverse (e.g., negative of one to positive of another), with the corresponding emf treated as negative.

12. Kirchhoff's Rules

• Developed by Gustav Robert Kirchhoff (1824–1887), a German physicist known for spectroscopy and contributions to mathematical physics.
• These rules are very useful for analyzing complex electric circuits that cannot be simplified by simple series/parallel combinations.
• When applying, assign a symbol (e.g., I) and a directed arrow for current in each resistor and source.

12.1 Junction Rule (Kirchhoff's First Rule / Current Law)

• Statement: At any junction in a circuit, the sum of currents entering the junction is equal to the sum of currents leaving the junction.
• Proof/Basis: This rule is a direct consequence of the conservation of electric charge. For steady currents, there is no accumulation of charges at any point or junction.

12.2 Loop Rule (Kirchhoff's Second Rule / Voltage Law)

• Statement: The algebraic sum of changes in potential around any closed loop involving resistors and cells is zero.
• Proof/Basis: This rule follows from the fact that electric potential is dependent only on the location. If you start at a point in a closed loop and return to the same point, the total change in potential must be zero.

13. Wheatstone Bridge

• A circuit arrangement consisting of four resistors (R₁, R₂, R₃, R₄).
• A source (battery arm) is connected across one pair of diagonally opposite points (e.g., A and C).
• A galvanometer (G, to detect currents, galvanometer arm) is connected between the other two vertices (e.g., B and D).
• Balanced Bridge Condition: Of special interest is the case where the resistors are configured such that the current through the galvanometer (I_g) is zero (null deflection).
• Balance Condition Formula: When the bridge is balanced (I_g = 0), the relationship between the resistors is: R₂/R₁ = R₄/R₃ or R₁/R₃ = R₂/R₄.
• Application: The Wheatstone bridge and its balance condition provide a practical method for determining an unknown resistance. If R₁, R₂, and R₃ are known, and R₄ is unknown, adjust R₃ until I_g = 0, then calculate R₄ = (R₂R₃)/R₁.
• A practical device using this principle is called the meter bridge.

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To visualize the concept of current and resistance, imagine a busy highway during rush hour.

• Electric current (I) is like the flow of cars on the highway – the number of cars passing a certain point per second.
• Voltage (V) is like the "push" or "gradient" that makes cars want to move from a higher elevation to a lower one, or the incentive for them to move along the highway.
• Resistance (R) is like the traffic itself – the factors that impede the flow of cars.
  • A long highway (length, l) will have more traffic jams and slowdowns, increasing resistance.
  • A narrow highway (small cross-sectional area, A) will also cause more congestion, increasing resistance.
  • The type of road surface or rules (resistivity, ρ) dictates how easily cars can move on that specific highway material.
• Drift velocity (v_d) is the average, slow progress of cars moving through dense traffic, even though individual cars might briefly accelerate and then stop (collide).
• Ohm's Law (V=IR) tells us that if you increase the "push" (voltage), more cars will flow (current), assuming the traffic conditions (resistance) remain the same. Alternatively, if the traffic conditions worsen (resistance increases) with the same "push," fewer cars will flow.
• Kirchhoff's Junction Rule is like saying that at any highway interchange, the total number of cars entering must equal the total number of cars leaving; cars don't just disappear or spontaneously appear.
• Kirchhoff's Loop Rule is like saying that if you start a journey at one point on the highway, drive around a complex loop, and return to your starting point, your net change in elevation or position from your starting point is zero. The sum of all ups and downs (potential changes) in a closed loop is zero.