CBSE Class 12 Physics Current Electricity Notes Set 04

1. Electric Current

Electric current is the rate of flow of charge through any cross-section of a conductor. If a net charge Δq crosses the section in time Δt, the average current is
I = \( \frac{\Delta q}{\Delta t} \), and the instantaneous current is I = \( \frac{dq}{dt} \).
1. SI unit: ampere (A); 1 A = 1 C s⁻¹. Smaller units: 1 mA = 10⁻³ A, 1 μA = 10⁻⁶ A.
2. The direction of conventional current is the direction in which positive charge would move, i.e. opposite to the motion of electrons.
3. Current has a direction but is a scalar: currents meeting at a point add algebraically, not by the law of vector addition.
4. If n electrons pass a section in time t, I = \( \frac{ne}{t} \).

2. Electric Currents in Conductors

In a metal, the outer electrons of the atoms are free to move through the lattice. With no electric field these free electrons move randomly in all directions, colliding with the fixed positive ions, so there is no net flow of charge in any direction and the current is zero.
When a potential difference is applied (for example by a cell), an electric field is set up inside the conductor. Each free electron then acquires a small average velocity opposite to the field, superposed on its random motion. This steady net flow constitutes the current. In electrolytes and ionised gases both positive and negative ions carry the current.

3. Ohm's Law

At constant temperature (and other physical conditions), the current through a metallic conductor is directly proportional to the potential difference across its ends.
V ∝ I  ⇒  V = IR, where R is the resistance of the conductor. SI unit: ohm (Ω); 1 Ω = 1 V A⁻¹.

Dependence on dimensions: For a uniform conductor of length l and cross-sectional area A,
R = \( \frac{\rho l}{A} \), where ρ is the resistivity (specific resistance) of the material. SI unit of ρ: Ω m.
1. Resistance depends on the material, the dimensions and the temperature. Resistivity depends only on the material and the temperature, not on the shape or size.
2. When a wire is stretched to n times its length (volume constant), its area becomes 1/n times, so R becomes n² times.

Current density and the vector form of Ohm's law: Current density is the current per unit area normal to the flow, j = \( \frac{I}{A} \) (unit A m⁻²); it is a vector along the direction of current.
Using V = El and R = ρl/A in V = IR gives E = ρj, or
\( \vec{j} = \sigma \vec{E} \), where σ = \( \frac{1}{\rho} \) is the conductivity (unit Ω⁻¹ m⁻¹ or S m⁻¹).

4. Drift of Electrons and the Origin of Resistivity

In an electric field E, each free electron experiences a force −eE and an acceleration
\( \vec{a} = -\frac{e\vec{E}}{m} \)
Between collisions the electron gains velocity; at each collision its velocity is randomised again. The average time between successive collisions is the relaxation time τ (about 10⁻¹⁴ s in metals). Averaging over all electrons, the random velocities cancel and the average velocity, called the drift velocity, is
\( \vec{v}_d = -\frac{e\vec{E}}{m}\tau \)

Relation between current and drift velocity: If n is the number of free electrons per unit volume and A is the area of cross-section, the charge crossing the area in time Δt is neAv_dΔt. So
I = neAv_d  and  j = nev_d

Expression for resistivity: Substituting v_d = eEτ/m in j = nev_d,
j = \( \frac{ne^2\tau}{m} E \)  ⇒  σ = \( \frac{ne^2\tau}{m} \)  and  ρ = \( \frac{m}{ne^2\tau} \)
This shows how resistivity arises from collisions: fewer free electrons (smaller n) or more frequent collisions (smaller τ) mean a larger resistivity.

Key points:
1. Drift speeds are very small, typically about 10⁻⁴ m s⁻¹ to 10⁻³ m s⁻¹, while random thermal speeds are about 10⁵ m s⁻¹.
2. A bulb still lights almost at once when switched on, because the electric field is set up throughout the circuit very quickly and the free electrons everywhere in the wire start drifting together.
3. For a steady current in a wire of non-uniform area, I is the same at every section, so v_d is larger where the wire is thinner.

5. Mobility

Mobility is the magnitude of the drift velocity per unit electric field.
μ = \( \frac{|v_d|}{E} \) = \( \frac{e\tau}{m} \)
SI unit: m² V⁻¹ s⁻¹. Mobility is positive for both electrons and holes. Conductivity can be written σ = neμ.

6. Limitations of Ohm's Law

Ohm's law is not a fundamental law of nature. Materials and devices that obey V = IR with constant R are called ohmic; others are non-ohmic. Ohm's law fails when:
1. V stops being proportional to I (the V–I graph is not a straight line).
2. The relation between V and I depends on the sign of V: reversing V changes the magnitude of the current, not only its direction (for example, a semiconductor diode).
3. The relation is not unique: more than one value of V gives the same current I (for example, gallium arsenide).

7. Resistivity of Various Materials

Materials are grouped by their resistivity:
1. Conductors (metals): low resistivity, about 10⁻⁸ Ω m to 10⁻⁶ Ω m (copper ≈ 1.7 × 10⁻⁸ Ω m, silver lowest).
2. Semiconductors (silicon, germanium): intermediate resistivity, which decreases as temperature rises and can be changed greatly by adding small amounts of impurities.
3. Insulators (glass, rubber, mica): very high resistivity, about 10¹⁴ Ω m or more.
Alloys such as nichrome, manganin and constantan have higher resistivity than pure metals and a resistivity that changes very little with temperature. Nichrome is used in heating elements; manganin and constantan are used for standard resistors.

8. Temperature Dependence of Resistivity

Over a limited range of temperature, the resistivity of a metallic conductor is given approximately by
ρ_T = ρ₀ [1 + α(T − T₀)]
where ρ₀ is the resistivity at a reference temperature T₀ and α is the temperature coefficient of resistivity (unit °C⁻¹ or K⁻¹). The same relation is used for resistance: R_T = R₀[1 + α(T − T₀)], so α = \( \frac{R_T - R_0}{R_0 (T - T_0)} \).

1. Metals: α is positive. As temperature rises, the ions vibrate more and collisions become more frequent, so τ decreases. n hardly changes, so ρ = m/(ne²τ) increases.
2. Alloys such as nichrome: ρ is high and increases only slightly with temperature (very small α).
3. Semiconductors and insulators: α is negative. As temperature rises, the number density n of charge carriers increases rapidly. This increase outweighs the decrease in τ, so ρ decreases.

9. Electrical Energy and Power

When a charge q moves through a potential difference V, the energy given up is qV. For a current I flowing for time t, the electrical energy converted (into heat in a resistor) is
W = VIt = I²Rt = \( \frac{V^2}{R} t \) (Joule heating)
Power: P = VI = I²R = \( \frac{V^2}{R} \). SI unit: watt (W). The commercial unit of energy is the kilowatt hour: 1 kWh = 3.6 × 10⁶ J.
1. For a device rated P watts at V volts, R = V²/P. Across a fixed supply, a lower-power bulb has the higher resistance.
2. Power transmission: To deliver power P through cables of resistance R_c at voltage V, the current is I = P/V and the power lost is P_c = I²R_c = \( \frac{P^2 R_c}{V^2} \). This is why power is transmitted at very high voltages and stepped down near the consumer.

10. Cells, EMF and Internal Resistance

A cell has two electrodes (positive P and negative N) dipped in an electrolyte. Chemical action keeps a potential difference between them.
1. EMF (ε): the potential difference between the terminals of the cell when no current is drawn from it (open circuit). It is the energy supplied by the cell per unit charge, and its unit is the volt.
2. Internal resistance (r): the resistance offered by the electrolyte to the flow of current inside the cell.

When the cell drives a current I through an external resistance R:
I = \( \frac{\varepsilon}{R + r} \)  and the terminal voltage is  V = IR = ε − Ir
1. V < ε while the cell supplies current (discharging); V = ε when I = 0.
2. While the cell is being charged, current enters at the positive terminal and V = ε + Ir, so V > ε.
3. The maximum current a cell can give is the short-circuit current I_max = ε/r (R = 0).
4. Internal resistance can be found from r = R \( \left( \frac{\varepsilon}{V} - 1 \right) \).

11. Cells in Series and in Parallel

(i) Series (negative of one cell joined to positive of the next):
ε_eq = ε₁ + ε₂ and r_eq = r₁ + r₂
If one cell is connected the wrong way round (opposing), ε_eq = ε₁ − ε₂ while r_eq is still r₁ + r₂.
For n identical cells in series: ε_eq = nε, r_eq = nr, and I = \( \frac{n\varepsilon}{R + nr} \).

(ii) Parallel (positive terminals joined together and negative terminals joined together):
ε_eq = \( \frac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2} \) and \( \frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} \)
For m identical cells in parallel: ε_eq = ε, r_eq = r/m, and I = \( \frac{\varepsilon}{R + r/m} \).

12. Kirchhoff's Rules

For circuits that cannot be reduced to simple combinations, two rules are used.
(i) Junction rule: At any junction, the sum of the currents entering is equal to the sum of the currents leaving, i.e. ΣI = 0 at a junction. It follows from the conservation of charge: charge cannot pile up at a junction in a steady state.
(ii) Loop rule: The algebraic sum of the changes in potential around any closed loop is zero, i.e. ΣΔV = 0 (equivalently Σε = ΣIR around the loop). It follows from the conservation of energy, because the electrostatic force is conservative.

Sign convention while going round a loop:
1. Across a resistor in the direction of the assumed current: potential change −IR; against the current: +IR.
2. Across a cell from the negative to the positive terminal: +ε; from positive to negative: −ε.
3. If a current comes out negative after solving, its actual direction is opposite to the one assumed.
Applying these rules gives the familiar results R = R₁ + R₂ for resistors in series and \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \) in parallel, which are useful when solving circuit problems.

13. Wheatstone Bridge

A Wheatstone bridge has four resistances P, Q, R and S joined to form a closed quadrilateral ABCD. A cell is connected across A and C and a galvanometer G across B and D.

The bridge is balanced when no current flows through the galvanometer (I_g = 0). Then B and D are at the same potential. Applying the loop rule to loops ABDA and BCDB with I_g = 0 gives
\( \frac{P}{Q} = \frac{R}{S} \) (balance condition)
1. If three of the resistances are known, the fourth can be found: S = \( \frac{QR}{P} \).
2. In a balanced bridge the galvanometer arm carries no current and can be removed while working out the equivalent resistance.
3. The balance is unaffected if the cell and the galvanometer are interchanged.

14. Quick Formula Recap