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APEX CLASS ESTD 2018

IITJEE: Wheatstone Bridge, Meter Bridge & RC Transient Circuits Lab

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Ashish Kumar Bahekar

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Apex class • IITJEE Current Electricity Lab

Wheatstone Bridge, Meter Bridge & RC Transient Circuits Lab

Master Null-Point Balance Conditions (P/Q = R/S) and RC Capacitor Shortcuts (t = 0, t = τ, t = ∞)!

🔬 Select a Bridge Circuit Configuration to Inspect Potentials & Null Point:
G=0 P = 4 Ω Q = 8 Ω R = 6 Ω S = 12 Ω A (12V) Node B (V_B = 8V) Node D (V_D = 8V) C (0V) ✅ Balanced Condition: P / Q = R / S 4 / 8 = 6 / 12 = 1 / 2 (or PS = QR = 48) V_B = V_D = 8V → Current I_G = 0 A! • Remove Central Arm BD Completely: R_eq = (P + Q) ∥ (R + S) = 12 Ω ∥ 18 Ω R_eq = (12 × 18) / (12 + 18) = 7.2 Ω! 🏆 JEE Trick: Swapping the Battery and Galvanometer in a balanced Wheatstone bridge does NOT affect the null point! I_G ↓ P = 4 Ω Q = 8 Ω R = 12 Ω S = 6 Ω Node B (Higher V_B) Node D (Lower V_D) ⚠️ Unbalanced Bridge (P/Q ≠ R/S) Here P/Q = 1/2, while R/S = 2/1 Since P/Q < R/S → Potential V_B > V_D! • Current flows from B → D through G • Solve using Nodal Analysis or Star-Delta! ⚡ Direction of Current Rule: If P·S < Q·R, then V_B > V_D (current flows B → D). If P·S > Q·R, then V_D > V_B (D → B)! R = 12 Ω S = 18 Ω (?) 0 cm 40 cm (Jockey J) 100 cm G=0 ← l = 40 cm → ← (100 – l) = 60 cm → 📏 Meter Bridge Formula: R / S = l / (100 – l) → 12 / S = 40 / 60 → Unknown S = 12 × (60 / 40) = 18.0 Ω! 1. Shunting Trap (Most Repeated PYQ) If a resistor is connected in PARALLEL with S: • Effective right-gap resistance S_eff DECREASES • Since l / (100-l) = R / S_eff, l INCREASES! (Null point shifts right toward 100 cm end) 2. End Corrections (α and β) Account for contact resistance at 0 & 100 cm ends: R / S = (l + α) / (100 – l + β) • Highest Sensitivity: When l = 50 cm (R = S) • Wire radius doubled → Null point UNCHANGED! 🎯 Experimental Physics Fact: If the radius of the uniform meter bridge wire is doubled, the balance length l does NOT change!
Null Condition
P / Q = R / S
V_B = V_D → I_G = 0 A
Meter Bridge Formula
S = R (100 – l) / l
l measured from Left Gap (R)
Max Accuracy Point
l = 50 cm (Midpoint)
Minimizes % Error ΔS/S
Wire Material Used
Manganin / Constantan
High ρ & Near-Zero α (Temp Coeff)
⏱️ Series RC Circuit (Time Constant τ = R·C, Max Charge Q₀ = CV). Select Time State:
At t = 0⁺ (Uncharged Capacitor q = 0) V R C acts as WIRE! (V_C = 0) ⚡ JEE Rule #1: At t = 0⁺, Replace C with a WIRE! • Initial Charge on C: q(0⁺) = 0 C • Potential Across C: V_C(0⁺) = q / C = 0 V • Initial Current is MAXIMUM: I₀ = V / R • All battery voltage drops across Resistor R! 🚀 Instantaneous Shortcut: The moment a switch closes, an uncharged capacitor offers ZERO resistance! Q₀ = CV Charge Growth: q(t) = Q₀(1 – e⁻ᵗ/ᵀ) t = τ = RC q(τ) = 0.632 Q₀ (63.2%!) Charging Current Decay: I(t) = (V/R) e⁻ᵗ/ᵀ t = τ = RC I(τ) = 0.368 I₀ (36.8%!) ⏱️ Time Constant τ = R_eq × C: At t = τ, Charge reaches 63.2% of Q₀ while Current drops to 36.8% (1/e) of I₀! At t → ∞ (Steady State: Fully Charged!) OPEN CIRCUIT! (I_C = 0 A) 🛑 JEE Rule #2: At t → ∞, Replace C with OPEN CIRCUIT! • Current through Capacitor Branch: I_C = 0 A • Final Charge on C: Q_max = C × V_branch • Stored Energy: U = ½ C V² | Heat in R = ½ C V² • Battery does W = CV² work (50% stored, 50% heat!) 💡 Multi-Loop RC Trick: In steady state, snip out the capacitor branch, find voltage V across its nodes, then Q = CV! Discharging: q(t) = Q₀ e⁻ᵗ/ᵀ & V(t) = V₀ e⁻ᵗ/ᵀ t_½ = 0.693 τ 50% Charge at t = τ ln(2) t = τ 36.8% Charge at t = τ Discharging Checkpoints • Charge: q(t) = Q₀ e⁻ᵗ/ᴿᶜ • Half-Charge Time: t = 0.693 R C • Stored Energy: U(t) = U₀ e⁻²ᵗ/ᴿᶜ • Energy drops to 50% twice as fast (0.346 RC)! ⚠️ JEE Trap: Because Energy U = q²/(2C), U decays as e⁻²ᵗ/ᵀ — its time constant is τ / 2 = RC / 2!
Capacitor Behavior
Short Circuit (Wire) 63.2% Charged Open Circuit (Broken) Source of Current
Circuit Reduction Rule
Charge on Capacitor q(t)
0 Coulombs 0.632 C V Q_max = C V Q₀ e⁻ᵗ/ᴿᶜ
Continuous Function of Time
Current in Branch I(t)
I_max = V / R 0.368 (V / R) 0 Amperes! -(Q₀ / RC) e⁻ᵗ/ᴿᶜ
Always Decays Exponentially
Time Constant (τ)
τ = R_thevenin × C
Dimensions of [R][C] = [Time T]
🔥 How to Find R_eq in Complex Multi-Resistor RC Circuits (Thevenin Trick):
1. Short-circuit all ideal voltage sources (replace batteries with plain wires).
2. Find equivalent resistance R_th across the two terminals where the capacitor is connected → τ = R_th × C!
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