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📑 In This Chapter Guide (Table of Contents)
1. Drift Velocity & Microscopic Derivation of Ohm’s Law
Drift Velocity (v_d): The average velocity with which free conduction electrons get drifted towards the positive terminal of a conductor under an applied electric field.
v_d = - (e · E · τ) / m
(where e = elementary charge, E = electric field, m = electron mass, τ = average relaxation time)
Relation Between Electric Current (I) and Drift Velocity (v_d):
I = n · e · A · v_d ⟹ Current Density: J = I / A = n · e · v_d
Derivation of Resistivity: Substituting v_d = (eEτ)/m:
I = n · e · A · [ (e · V · τ) / (m · L) ] ⟹ V / I = [ m / (n · e² · τ) ] · (L / A)
Comparing with R = ρ(L/A), we obtain electrical resistivity: ρ = m / (n · e² · τ).
2. Kirchhoff’s Circuit Laws & Sign Conventions
- Kirchhoff’s First Rule (Junction Rule / Current Law - KCL): In any electrical network, the algebraic sum of currents meeting at any junction is zero:
∑ I = 0. (Sum of currents entering a junction = Sum of currents leaving). Represents Conservation of Electric Charge. - Kirchhoff’s Second Rule (Loop Rule / Voltage Law - KVL): The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero:
∑ ΔV = ∑ E - ∑ (I·R) = 0. Represents Conservation of Energy.
3. Wheatstone Bridge Balanced Condition Derivation
A Wheatstone bridge consists of four resistors P, Q, R, and S arranged in a quadrilateral ABCD with a galvanometer G connected between B and D:
Under balanced condition, no current flows through the galvanometer: I_g = 0 ⟹ V_B = V_D.
Applying Loop Rule to Loop ABDA: -I₁·P + I₂·R = 0 ⟹ I₁·P = I₂·R ... (Equation 1)
Applying Loop Rule to Loop BCDB: -I₁·Q + I₂·S = 0 ⟹ I₁·Q = I₂·S ... (Equation 2)
Dividing Equation 1 by Equation 2: (I₁·P) / (I₁·Q) = (I₂·R) / (I₂·S) ⟹ P / Q = R / S
This is the famous Wheatstone Bridge Balance Condition. Used in Meter Bridge experiments to accurately determine unknown resistance.
💡 Frequently Asked Questions (FAQ)
❓ Why does the resistance of a metallic conductor increase with rising temperature?
Electrical resistance is inversely proportional to relaxation time: R ∝ 1/τ. When temperature increases, metallic positive ions vibrate with greater thermal amplitude, increasing the collision frequency of free conduction electrons. This sharply reduces the average relaxation time (τ), thereby increasing resistance and resistivity.
❓ Differentiate between EMF and Terminal Potential Difference of a battery cell.
Electromotive Force (EMF, E) is the maximum potential difference between the electrodes of a cell in an open circuit when no current is drawn. Terminal Potential Difference (V) is the voltage across the electrodes when the cell is discharging through an external circuit: V = E - I·r (where r is internal resistance).
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