Govt. Exams
Both particles gain same kinetic energy, so mv²/2 is same. Since r = mv/eB and proton has much larger mass than electron, proton has larger radius of curvature.
Using Ampere's law for a long straight wire, ∮B·dl = μ₀I_enclosed. For a circular path of radius r: B(2πr) = μ₀I, giving B = μ₀I/2πr.
The magnetic moment of an electron in the first Bohr orbit equals 1 Bohr magneton (μ_B = eℏ/2m_e ≈ 9.27 × 10⁻²⁴ J/T). This is a fundamental quantum result.
Torque on a current loop in a magnetic field is τ = NIAB sin(θ), where A is the area and θ is the angle. Maximum torque occurs when sin(θ) = 1, giving τ_max = BIab for N=1.
Currents in opposite directions repel each other. The force per unit length is F/ℓ = μ₀I₁I₂/2πd, making total force F = μ₀I₁I₂ℓ/2πd (repulsive).
For a charged particle in a magnetic field, centripetal force equals magnetic force: mv²/r = evB, giving r = mv/eB. This is the radius of the circular path.
Resistance of element R = V²/P = 220²/1000 = 48.4Ω (constant). At 110V: P' = V'²/R = 110²/48.4 ≈ 250W. Power varies with square of voltage.
When stretched, length becomes 1.5L. Volume remains constant (LA = A'L'), so A' = A/1.5. New resistance R' = ρ(1.5L)/(A/1.5) = 2.25ρL/A = 2.25R
P = V²/R. New P = (2V)²/(R/2) = 4V²×2/R = 8(V²/R) = 8P₀
Tungsten filament's resistance increases significantly with temperature due to heating, causing non-linear I-V characteristic (non-ohmic).
