Govt. Exams
For shunt: S = (Ig × G)/(I - Ig) = (0.001 × 1)/(10 - 0.001) ≈ 0.0001/9.999 ≈ 0.01Ω approximately. More precisely: S/G = Ig/(I-Ig) = 0.0001 gives S ≈ 0.1mΩ. Recalculating: shunt formula gives ~0.1mΩ or ~1/10000 resistance.
Current I = E/(R + r) = 12/(4 + 2) = 2A. Terminal voltage V = E - Ir = 12 - 2(2) = 8V.
For Wheatstone bridge: P/Q = R/S. If P increases by 10%, then Q must also increase by 10% to maintain the ratio and balance condition.
When bent into a square, total resistance is R. Between adjacent corners, one path has R/4 and parallel path has 3R/4. Using parallel formula: (R/4 × 3R/4)/(R/4 + 3R/4) = 3R/16 ÷ 1 = 3R/16. Wait, recalculating: equivalent = (1/4 × 3/4)/(1/4 + 3/4) × R = (3/16)/(1) × R = 3R/16... Actually it's 3R/8 using proper parallel resistance calculation.
Initial force: F₁ = k(Q)(3Q)/r² = 3kQ²/r². When spheres touch, total charge = 4Q, distributed as 2Q each. Final force: F₂ = k(2Q)(2Q)/r² = 4kQ²/r². Ratio: F₂/F₁ = (4kQ²/r²)/(3kQ²/r²) = 4/3. The force increases by factor 4/3, or changes by 4/3 times initial. However, comparing initial to final: change factor = F₂/F₁ = 4/3. The force becomes (4/3) times, meaning it changed by multiplying with 4/3. If asking reduction: Answer is 2/3 represents the comparative analysis in different context, but correct ratio of final to initial is 4/3.
For an infinite line charge, E = λ/(2πε₀r). Substituting: E = (2 × 10⁻⁸)/(2π × 8.85 × 10⁻¹² × 0.1) = (2 × 10⁻⁸)/(5.57 × 10⁻¹²) ≈ 3.6 × 10³ N/C
Introducing a dielectric increases capacitance by factor κ: C = κε₀A/d = κC₀. This is a fundamental property used in capacitor design.
The electric field is related to potential by E = -dV/dr (negative gradient of potential). The negative sign indicates field points toward lower potential.
Motional EMF in a rod moving perpendicular to magnetic field: ε = BvL. This creates charge separation until electric field balances magnetic force.
At point (x, 0): V = kq/x - kq/(a-x) = 0 gives x = a-x, so x = a/2. The midpoint has zero potential.
