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Physics 212 Lecture 17 Faraday’s Law Physics 212 Lecture 17, Slide 1 Motional EMF Change Area of loop Change magnetic field through loop Change orientation of loop relative to B In each case the flux of the magnetic field through the circuit changes with time and an EMF is produced. B B dA d B dt EMF Physics 212 Lecture 17, Slide 2 Rotate the loop,change flux, generate emf. B Physics 212 Lecture 17, Slide 3 Move loop to a place wherethe B field is different, change flux, generate emf. B1 B2 v Physics 212 Lecture 17, Slide 4 Checkpoint 1a A copper loop is placed in a uniform magnetic field as shown. You are looking from the right. Suppose the loop is moving to the right. The current induced in the loop is: A. zero B. clockwise C. counterclockwise • Motional emf is ZERO •vXB=0 • no charge separation • no E field • no emf • The flux is NOT changing • B does not change • the area does not change • the orientation of B and A does not change Physics 212 Lecture 17, Slide 5 Checkpoint 1c Now suppose that the loop is spun around a vertical axis as shown, and that it makes one complete revolution every second. The current induced in the loop: A. Is zero B. Changes direction once per second C. Changes direction twice per second Current changes direction every time the loop becomes perpendicular with the B field emf ~ d/dt (B dA = max) d/dt (B dA ) = 0 O X B dA O B X dA Physics 212 Lecture 17, Slide 6 Faraday’s Discovery d B emf dt True no matter how we change the flux. In fact, the circuit may be stationary (no motional EMF) and only the B-field changes with time. An EMF is still produced. This implies that: d B emf E dl dt Faraday’s Law Change the B field in time so flux changes. Induce an emf nnd therefore an Electric field. This emf tries to oppose the change in flux. (Lenz’s Law) d B emf E d dt B(t) decreasing Induces an E field even if there is no circuit there! Physics 212 Lecture 17, Slide 8 Checkpoint 1b A copper loop is placed in a uniform magnetic field as shown. You are looking from the right. Looking from right X X X X X X X X X X X X X X X X Now suppose the that loop is stationary and thatCheckpoint the magnetic field 1b is decreasing in time. The current induced in the loop is: A. zero B. clockwise C. counterclockwise • Motional emf is ZERO • Circuit is stationary ! • HOWEVER: The flux is changing • B decreases in time • current induced to oppose the flux change • clockwise current tries to restore B that was removed X X X X X X X X X X X X X X X X X X X X X X X X Clockwise current tries to restore B Physics 212 Lecture 17, Slide 9 Checkpoint 2 A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet F O X B B Like poles repel (copper is not ferromagnetic) Ftotal < mg Will the acceleration a of the falling ring in the presence of the magnet be any different than it would have been under the influence of just gravity (i.e. g)? A. a > g B. a = g C. a < g a<g This one is hard ! B field increases upward as loop falls Clockwise current (viewed from top) is induced Physics 212 Lecture 17, Slide 10 Checkpoint 2 A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet HOW IT WORKS Looking down B (copper is not ferromagnetic) Will the acceleration a of the falling ring in the presence of the magnet be any different than it would have been under the influence of just gravity (i.e. g)? A. a > g B. a = g C. a < g This one is hard ! B field increases upward as loop falls Clockwise current (viewed from top) is induced Main Field produces horizontal forces “Fringe” Field produces vertical force I I B IL X B points UP Ftotal < mg a<g Physics 212 Lecture 17, Slide 11 Calculation A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. y a v0 B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x What is the direction and the magnitude of the force on the loop when half of it is in the field? • Conceptual Analysis – – Once loop enters B field region, flux will be changing in time Faraday’s Law then says emf will be induced • Strategic Analysis – – – Find the emf Find the current in the loop Find the force on the current Physics 212 Lecture 17, Slide 12 Calculation A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. emf d B dt y a v0 B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x What is the magnitude of the emf induced in the loop just after it enters the field? (A) e = Babv02 (B) e = ½ Bav0 (C) e = ½ Bbv0 (D) e = Bav0 (E) e = Bbv0 y a v0 B x x x x x x x b x x x x x x x a x x x x x x x x x x x x x x Change in Flux = dB = BdA = Bav0dt x In a time dt it moves by v0dt The area in field changes by dA = v0dt a d B Bav o dt Physics 212 Lecture 17, Slide 13 Calculation emf A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. y a v0 d B dt B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x What is the direction of the current induced in the loop just after it enters the field? (A) clockwise (B) counterclockwise (C) no current is induced emf is induced in direction to oppose the change in flux that produced it y a v0 B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x Flux is increasing into the screen Induced emf produces flux out of screen x Physics 212 Lecture 17, Slide 14 Calculation A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. emf y a v0 d B dt B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x What is the direction of the net force on the loop just after it enters the field? (A) +y (B) -y (C) +x (D) -x Force on a current in a magnetic field: F IL B y b a B x x x x x x x v0 I x x x x x x x • Force on top and bottom segments cancel (red arrows) • Force on right segment is directed in –x direction. x Physics 212 Lecture 17, Slide 15 Calculation A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. What is the magnitude of the net force on the loop just after it enters the field? (A) F 4aBvo R (B) F a 2 Bvo R (C) F a 2 B 2vo2 / R F IL B emf y a v0 d B dt B b x x x x x x x x x x x x x x x x x x x x x x x x x x x x x F IL B e = Bav0 (D) F a 2 B 2vo / R F ILB since L B y b a F B x x x x x x x v0 I e Bavo I R R x x x x x x x B 2 a 2vo Bav o F aB R R ILB x Physics 212 Lecture 17, Slide 16 Follow-Up A rectangular loop (sides = a,b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in a the –z direction. t = dt: e = Bav0 y b v0 B x x x x x x x x x x x x x x x x x x x x x x x x x x x x x What is the velocity of the loop when half of it is in the field? Which of these plots best represents the velocity as a function of time as the loop moves form entering the field to halfway through ? (A) (B) (C) D) (E) X This is not obvious, but we know v must decrease Why? X b a Fright B x x x x x x x v0 I x x x x x x x X Fright points to left Acceleration negative Speed must decrease Physics 212 Lecture 17, Slide 17 Follow-Up A rectangular loop (sides = a,b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. y b a What is the velocity of the loop when half of it is in the field? v0 B x x x x x x x x x x x x x x x x x x x x x x x x x x x x e = Bav0 x Which of these plots best represents the velocity as a function of time as the loop moves form entering the field to halfway through ? dv (A) F a B v / R (D) m • Why (D), not (A)? 2 2 dt – F is not constant, depends on v a 2 B 2v dv F m R dt Challenge: Look at energy v vo e t 2 2 a where B mR Claim: The decrease in kinetic energy of loop is equal to the energy dissipated as heat in the resistor. Can you verify?? Physics 212 Lecture 17, Slide 18