emf induced in a moving conductor
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emf induced in a moving conductor conducting rod × × × × × × B × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ++ v Rod sliding on rails B + v B θ L v Magnetic force on a plus charge in the rod is θ F = q vB sinθ The work done by the field to move a + charge through the rod from one rail to the other is θ W = FL = q vB L sinθ The emf is the work per unit charge W E = = vBLsinθ θ q Magnetic Flux B B⊥ Φ = B ⊥A Φ is proportional to the number of magnetic field lines passing through the area. Units: weber (Wb) 1 Wb = 1 T• m2 Faraday’s Law ∆Φ dΦ E = → ∆t dt Disregard sign. We will use Lenz’s law to determine the polarity (+ and - terminals of the emf) EXAMPLE: The rod sliding with speed v on parallel rails. B + v B θ L v View the rails from directly above s+v ∆t s × × × × × × × × × × × × × × × × × × × × × × × L× × × × × v× × × ×v × × × × × × × × × × × × × × × × × × × × × × × Φ0 = B sinθ θ Ls Φf = B sinθ θ L (s + v ∆ t) ∆ Φ = Φf - Φ0 = B sinθ θ Lv∆t ∆Φ θ E = ∆ t = B L v sinθ as before. E = dΦ dt Φ = B ⊥A An emf is generated whenever Φ changes. 1) Change the shape of the loop. 2) Change the orientation of the loop. 3) Change the magnitude of B. Polarity of the induced emf Lenz’s law: The direction of an induced current (emf) is such as to oppose the cause producing it. Procedure for applying Lenz’s law: 1) Determine whether the magnetic flux is increasing or decreasing. 2) If the magnetic flux is increasing (decreasing), then the induced magnetic field will be opposite to (in the same direction as) the applied field. 3) Use the right hand rule to determine the direction of the induced current required to produce this induced magnetic field. EXAMPLE (Lenz’s Law): Ring A - Ring B + E R Determine the direction of the induced current in ring B when a) E changes b) R changes c) Ring B is moved to the right (left) Induced emf in a rotating coil W/2 B v= ωW/2 ϕ=ω ωt E one side = B L v sin(ϕ ϕ ) = B L v sin(ω ωt) The total emf is E = 2 B L v sin(ω ω t) ω t) = B L W ω sin(ω E = B A ω sin(ω ω t) ω=2πf For residential electric power we generally have f=60 Hz (i.e. 60 cycles per second) +BAω 1.00 0.00 -1.00 -BAω t Another approach: Φ = B A cos(ω ω t) At t=0, the plane of the loop is perpendicular to the magnetic field and Φ = B A dΦ E = dt E = B A ω sin(ω ω t) as before
