Class 12 Physics Notes Chapter 3 (Current electricity) – Physics Part-I Book
Alright class, let's delve into Chapter 3: Current Electricity. This is a fundamental chapter for understanding how charges flow and how we analyze electrical circuits, crucial for many government exams involving Physics.
Chapter 3: Current Electricity - Detailed Notes
1. Electric Current (I)
- Definition: The rate of flow of electric charge through any cross-section of a conductor.
- Formula: If charge Δq flows in time Δt, the average current I_avg = Δq/Δt.
The instantaneous current I = dq/dt. - Nature: A scalar quantity, although we assign a direction (conventional current).
- Direction (Conventional Current): The direction of flow of positive charge (or opposite to the direction of flow of electrons).
- Unit: SI unit is Ampere (A). 1 A = 1 C/s.
- Current Carriers: In metallic conductors - free electrons; In electrolytes - positive and negative ions; In semiconductors - electrons and holes.
2. Electric Current in Conductors & Drift Velocity (v_d)
- Mechanism: In a conductor without an electric field, free electrons move randomly (like gas molecules), and their average thermal velocity is zero. Net current is zero.
- With Electric Field (E): When an electric field is applied, electrons experience a force (F = -eE) and accelerate. However, they collide frequently with ions/atoms in the conductor.
- Drift Velocity (v_d): The average velocity with which free electrons get drifted towards the positive end of the conductor under the influence of an external electric field. It's very small (order of 10⁻⁴ m/s).
- Relaxation Time (τ): The average time interval between two successive collisions of an electron with the ions/atoms.
- Relation: v_d = (eE/m)τ , where e = charge of electron, m = mass of electron.
- Relation between Current and Drift Velocity:
Consider a conductor of length l, area A, with electron density n (number of free electrons per unit volume).
Total charge in length l = (nAl)e
Time taken to cross length l = t = l/v_d
Current I = Charge/Time = (nAl)e / (l/v_d) = I = n e A v_d - Current Density (J): Current per unit area of cross-section, taken normal to the current flow.
J = I/A = n e v_d
It's a vector quantity: J = n e v_d (Note: v_d is opposite to E for electrons, so J = -n e v_d. But conventionally, J is in the direction of E, so we use magnitude relation or consider charge q: J = nq v_d).
Also, J = σ E (Vector form of Ohm's Law).
3. Ohm's Law
- Statement: Provided the physical conditions (like temperature, pressure) remain unchanged, the current (I) flowing through a conductor is directly proportional to the potential difference (V) across its ends.
- Formula: V ∝ I => V = IR
- Resistance (R): The constant of proportionality. It is the opposition offered by the conductor to the flow of current.
R = V/I - Unit of Resistance: Ohm (Ω). 1 Ω = 1 V/A.
- V-I Graph: For ohmic conductors (obeying Ohm's Law), the V-I graph is a straight line passing through the origin. The slope of V-I graph gives Resistance (R), and the slope of I-V graph gives Conductance (G = 1/R).
4. Resistivity (ρ) and Conductivity (σ)
- Factors Affecting Resistance:
- Length (l): R ∝ l
- Area of cross-section (A): R ∝ 1/A
- Nature of material
- Temperature
- Formula: R = ρ (l/A)
- Resistivity (ρ) or Specific Resistance: Resistance offered by a material per unit length for a unit cross-sectional area. It depends only on the nature of the material and temperature.
ρ = RA/l - Unit of Resistivity: Ohm-meter (Ω m).
- Conductivity (σ): The reciprocal of resistivity. It measures how easily a material allows current to flow.
σ = 1/ρ - Unit of Conductivity: Siemens per meter (S/m) or (Ω m)⁻¹ or mho/m.
- Relation from Drift Velocity: We derived R = ml / (ne²τA). Comparing with R = ρl/A, we get:
ρ = m / (ne²τ)
σ = ne²τ / m
5. Mobility (μ)
- Definition: Magnitude of the drift velocity per unit electric field.
μ = |v_d| / E - Formula: Using v_d = (eE/m)τ, we get μ = (e/m)τ
- Unit: m² V⁻¹ s⁻¹
6. Temperature Dependence of Resistivity
- Conductors: Resistivity increases with temperature. As T increases, collisions become more frequent, so relaxation time (τ) decreases, increasing ρ (ρ ∝ 1/τ).
Approximate relation: ρ_T = ρ₀ [1 + α(T - T₀)] or R_T = R₀ [1 + α(T - T₀)]
where α is the temperature coefficient of resistance. For metals, α is positive. - Semiconductors & Insulators: Resistivity decreases with temperature. As T increases, the number density of charge carriers (n for electrons, p for holes) increases significantly, outweighing the decrease in τ. So, ρ decreases (ρ ∝ 1/n). α is negative.
- Alloys: Materials like Nichrome, Manganin, Constantan have high resistivity and low temperature coefficient of resistance. Used for making standard resistors, heating elements.
7. Limitations of Ohm's Law
- Ohm's law is not a fundamental law. It holds for metallic conductors at constant temperature.
- Devices that do not obey Ohm's law are called non-ohmic devices (e.g., semiconductor diodes, transistors, vacuum tubes, electrolytes).
- Their V-I graphs are non-linear.
8. Electrical Energy and Power
- Electrical Energy (W or E): Work done by the source to maintain current in a circuit.
W = Vq = V(It) = I²Rt = (V²/R)t - Unit: Joule (J). Commercial unit: kilowatt-hour (kWh) or Board of Trade (BOT) unit.
1 kWh = (1000 W) x (3600 s) = 3.6 x 10⁶ J. - Electrical Power (P): The rate at which electrical energy is consumed or dissipated.
P = W/t = VI = I²R = V²/R - Unit: Watt (W). 1 W = 1 J/s.
- Joule's Law of Heating: Heat produced (H) in a resistor R carrying current I for time t is H = I²Rt.
9. Combination of Resistors
- Series Combination:
- Resistors connected end-to-end.
- Same current flows through all resistors.
- Voltage divides across resistors (V = V₁ + V₂ + ...).
- Equivalent Resistance: R_s = R₁ + R₂ + R₃ + ... (R_s is greater than the largest individual resistance).
- Parallel Combination:
- Resistors connected between the same two points.
- Same voltage across all resistors.
- Current divides through resistors (I = I₁ + I₂ + ...).
- Equivalent Resistance: 1/R_p = 1/R₁ + 1/R₂ + 1/R₃ + ... (R_p is smaller than the smallest individual resistance).
- For two resistors: R_p = (R₁ R₂) / (R₁ + R₂)
10. Cells, EMF (E), and Internal Resistance (r)
- Electromotive Force (EMF - E): The potential difference across the terminals of a cell when no current is drawn from it (open circuit). It represents the maximum potential difference the cell can provide. It's the work done by the non-electrostatic force (chemical force in the cell) per unit charge to move it from the lower potential terminal to the higher potential terminal within the cell. Unit: Volt (V).
- Internal Resistance (r): The opposition offered by the electrolyte and electrodes of the cell to the flow of current through the cell. Unit: Ohm (Ω).
- Terminal Potential Difference (V): The potential difference across the terminals of a cell when current is being drawn from it (closed circuit).
- Relationship (Discharging Cell): When a cell of EMF E and internal resistance r drives a current I through an external resistance R:
Total Resistance = R + r
Current I = E / (R + r)
Potential drop across R is V = IR.
Potential drop across internal resistance r is Ir.
Terminal Voltage V = E - Ir - Relationship (Charging Cell): When a cell is being charged by an external source providing current I:
Terminal Voltage V = E + Ir
11. Combination of Cells
- Series Combination:
- Cells connected end-to-end (negative of one to positive of next).
- Equivalent EMF: E_eq = E₁ + E₂ + ... + E_n (if aiding)
- Equivalent Internal Resistance: r_eq = r₁ + r₂ + ... + r_n
- Current: I = nE / (R + nr) (for n identical cells)
- Useful when external resistance R >> nr.
- Parallel Combination:
- All positive terminals connected together, all negative terminals connected together.
- Equivalent EMF (for identical cells): E_eq = E
- Equivalent Internal Resistance (for n identical cells): r_eq = r/n
- Current: I = E / (R + r/n) = nE / (nR + r) (for n identical cells)
- Useful when external resistance R << r/n.
- Mixed Grouping: Combination of series and parallel. Current is maximum when external resistance R equals the total internal resistance of the grouping.
12. Kirchhoff's Laws (For analyzing complex circuits)
- Kirchhoff's First Law (Junction Rule or KCL - Kirchhoff's Current Law): The algebraic sum of currents meeting at any junction in an electrical circuit is zero.
ΣI = 0
(Convention: Currents entering junction are positive, currents leaving are negative, or vice-versa).
Basis: Conservation of Charge. - Kirchhoff's Second Law (Loop Rule or KVL - Kirchhoff's Voltage Law): The algebraic sum of changes in potential around any closed loop involving resistors and cells in the circuit is zero.
ΣΔV = 0
(Convention:- Potential drop across resistor (-IR) if traversing in direction of current.
- Potential gain across resistor (+IR) if traversing opposite to current.
- Potential gain (+E) if traversing from negative to positive terminal of cell.
- Potential drop (-E) if traversing from positive to negative terminal of cell.)
Basis: Conservation of Energy.
13. Wheatstone Bridge
- Arrangement: Four resistors (P, Q, R, S) connected to form a quadrilateral. A galvanometer (G) is connected between one pair of opposite junctions, and a cell between the other pair.
- Principle: Used to find an unknown resistance accurately by comparison.
- Balanced Condition: When the potential difference across the galvanometer is zero (V_B = V_D), no current flows through it (I_g = 0). The bridge is said to be balanced.
In balanced condition: P/Q = R/S
(P, Q are ratio arms; R is known resistance; S is unknown resistance).
14. Metre Bridge (Slide Wire Bridge)
- Practical Form: A practical application of the Wheatstone bridge.
- Construction: Consists of a 1-meter long uniform wire (usually Manganin or Constantan) stretched along a scale, with copper strips, gaps for known resistance (R) and unknown resistance (S). A galvanometer is connected via a jockey.
- Working: The jockey is moved along the wire to find the balancing point (null deflection in galvanometer). Let the balancing length from one end (say, connected to R) be l₁. The remaining length is (100 - l₁).
- Formula: Applying Wheatstone principle: R / (Resistance of length l₁) = S / (Resistance of length (100 - l₁)).
Since wire is uniform, resistance ∝ length.
R / l₁ = S / (100 - l₁) or S = R (100 - l₁) / l₁ - Sources of Error: Non-uniformity of wire, end resistances (contact resistances at copper strips), heating effects. Minimized by swapping R and S and taking the average, using jockey gently.
15. Potentiometer
- Device: An instrument used to measure potential difference (or EMF) accurately without drawing any current from the source being measured.
- Principle: The potential drop (V) across any portion of a uniform wire carrying a constant current (I) is directly proportional to the length (l) of that portion.
V = (Potential Gradient) x l => V ∝ l
Potential Gradient (k): Potential drop per unit length of the potentiometer wire. k = V_wire / L_wire (Volt/meter). - Construction: A long uniform wire (several meters, often arranged in parallel segments) stretched on a board, connected to a driver cell (or battery) through a rheostat (for current control) and a key.
- Applications:
- Comparison of EMFs of Two Cells:
Balance the first cell (E₁) at length l₁. E₁ = k l₁.
Balance the second cell (E₂) at length l₂. E₂ = k l₂.
Ratio: E₁ / E₂ = l₁ / l₂ - Measurement of Internal Resistance of a Cell (r):
Balance the cell (E) alone at length l₁. E = k l₁.
Connect a known resistance (R) across the cell via a key. Balance the terminal voltage (V) across R at length l₂. V = k l₂.
We know V = E - Ir = E R / (R + r).
So, k l₂ = (k l₁) R / (R + r)
l₂ / l₁ = R / (R + r) => l₂(R + r) = l₁R => l₂r = (l₁ - l₂)R
Internal Resistance: r = R (l₁ / l₂ - 1)
- Comparison of EMFs of Two Cells:
- Superiority over Voltmeter: A potentiometer draws no current from the source at the balance point, hence it measures the actual EMF or potential difference accurately. A voltmeter always draws some current, so it measures V = E - Ir, which is less than E.
Practice MCQs
-
The SI unit of electrical conductivity is:
a) Ohm (Ω)
b) Ohm-meter (Ω m)
c) Siemens per meter (S/m)
d) Ampere (A) -
The drift velocity (v_d) of electrons in a conductor is related to the electric field (E) and relaxation time (τ) as:
a) v_d = (mE)/(eτ)
b) v_d = (eEτ)/m
c) v_d = (emτ)/E
d) v_d = Eτ/(em) -
For semiconductors, the temperature coefficient of resistance (α) is:
a) Positive
b) Negative
c) Zero
d) Infinite -
Kirchhoff's junction rule (KCL) is based on the conservation of:
a) Energy
b) Momentum
c) Charge
d) Mass -
Three resistors of 2 Ω, 3 Ω, and 6 Ω are connected in parallel. The equivalent resistance is:
a) 11 Ω
b) 1 Ω
c) 6 Ω
d) 0.5 Ω -
A cell of EMF 'E' and internal resistance 'r' is connected across an external resistance 'R'. The terminal potential difference 'V' across the cell is given by:
a) V = E + Ir
b) V = E / (R+r)
c) V = E - Ir
d) V = E -
In a balanced Wheatstone bridge, if the positions of the cell and galvanometer are interchanged, the balance condition will:
a) Change
b) Remain unchanged
c) Become E/G = R/S
d) Depend on the resistance values -
The principle of a potentiometer states that the potential drop across any portion of the uniform wire is directly proportional to its:
a) Resistance
b) Area of cross-section
c) Current
d) Length -
A wire of resistance R is stretched to double its length. Assuming the volume remains constant, its new resistance will be:
a) R/2
b) R
c) 2R
d) 4R -
Which of the following devices does NOT obey Ohm's Law?
a) Copper wire at constant temperature
b) Manganin wire
c) Semiconductor diode
d) Nichrome wire
Answers to MCQs:
- (c)
- (b)
- (b)
- (c)
- (b) [1/Rp = 1/2 + 1/3 + 1/6 = (3+2+1)/6 = 6/6 = 1 => Rp = 1 Ω]
- (c)
- (b)
- (d)
- (d) [l' = 2l. Volume V = Al = constant. A'l' = Al => A'(2l) = Al => A' = A/2. R' = ρ(l'/A') = ρ(2l)/(A/2) = 4 (ρl/A) = 4R]
- (c)
Remember to thoroughly understand the concepts behind these formulas and practice applying them to various problems. Good luck with your preparation!