University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

232

Pages:

6
Author:

Jabin F.

Current is flowing in infinite length wire:- ∮_C A⃗ · d⃗l = ∫_S B⃗ · d⃗s = magnetic flux (Φm) Ques. 1 Page-50 B_in = μ₀nI ẑ B_out = 0 Inside: Φm = ∫_S B⃗ · d⃗s = ∫₀^r ∫₀^(2π) μ₀nI r dr dφ = μ₀nIπr² Φm = μ₀nIπr² ⇒ ∮_C A⃗ · d⃗l = A_in 2πr = μ₀nIπr² ⇒ A_in = (μ₀nIπr²) / (2πr) = (μ₀nI) / 2 A_in α r # Outside: Φm = ∫_S B⃗ · d⃗s = μ₀nIπa² ⇒ ∮_C A_out · d⃗l = μ₀nIπa² ⇒ ∮_C A⃗ · d⃗l = A_in 2πr = μ₀nIπr² ⇒ A_in = (μ₀nIπr²) / (2πr) = (μ₀nI) / 2 A_in α r # Outside: Φm = ∫_S B⃗ · d⃗s = μ₀nIπa² ⇒ ∮_C A_out · d⃗l = μ₀nIπa² ⇒ A_out = (μ₀nIπa²) / (2πr) = (μ₀nI) / 2 A_out α 1/r Ques. 60 Page-48 E = E₀ĉ B = Bk̂ (no electric field) mv²/R = 2qvB R = mv / (2qB) if +q at a point: qv = (1/2)mv² q(EL) = (1/2)mv² ⇒ q/m = (v²) / (2EL) from eqs. (i) and (ii) V = (q/m)BR V = (v²/2EL)BR ⇒ V = (2EL) / BR Ques. 4 Page-48 (a) Inside: Ienc = 0, Bin = 0 Outside: Ienc = I surface current density (K) = I/(2πa) Ienc = ∫ K dl = ∫₀^(2π) K a dφ = Ka(2π) = I ∮ B⃗ · d⃗l = μ₀Ienc Bout 2πr = μ₀I Bout = (μ₀I) / (2πr) B_out = (μ₀I) / (2πs) φ̂ → similar to infinite wire. (b) J α S ⇒ J = csẑ, c = Constant. J = { csẑ, for s ≤ a 0, for s > a Total Current in the wire (dl_s, dl_φ, dl_z) I = ∫_S J ds_l = ∫₀^a ∫₀^(2π) cs dz s dφ I = c(2π) (a³/3) = (2πca³) / 3 ⇒ c = (3I) / (2πa³) J = (3IS) / (2πa³) ẑ given # inside Ienc = ∫_S J ds_l ⇒ inside amperian loop Ienc = ∫₀^s ∫₀^(2π) (3IS) / (2πa³) s ds dφ Ienc = (IS³) / (a³) Ienc < I ∮ B⃗ · d⃗l = μ₀Ienc B_in 2πTS = (μ₀IS³) / (a³) ⇒ B_in = (μ₀IS²) / (2πa³) φ̂ # outside Ienc = ? Ienc = ∫₀^a ∫₀^(2π) (IS³) / (a³) s ds dφ Ienc = I ∮ B⃗ · d⃗l = μ₀I B_out 2πs = μ₀I ⇒ B_out = (μ₀I) / (2πs) φ̂ similar to infinite wire.