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Quiz lecture 17 The figure shows an emf L induced in a coil. Which of the following can describe the current through the coil: L 1. constant and rightward 2. constant and leftward 3. increasing and rightward 4. decreasing and rightward 5. increasing and leftward 6. decreasing and leftward 3 & 6 1 & 6 3 & 4 4 & 5 2 & 4 11/13/2018

Energy in Magnetic Fields LECTURE 18 RL Circuits Calculate inductance Energy in Magnetic Fields

RL Circuits R I  a b L At t=0, the switch is closed and the current I starts to flow. Initially, an inductor acts to oppose changes in current through it. A long time later, it acts like an ordinary connecting wire. 11/13/2018

RL Circuits L I R  Find the current as a function of time. b L Find the current as a function of time. What about potential differences? 11/13/2018

Voltage on L RL Circuits ( on) Current Max =  37% Max at t=L/R Max = /R 63% Max at t=L/R I t Voltage on L Max =  37% Max at t=L/R  VL t 11/13/2018

RL Circuits L I R  Why does RL increase for larger L? b L Why does RL increase for larger L? Why does RL decrease for larger R? 11/13/2018

RL Circuits R I a b L After the switch has been in position for a long time, redefined to be t=0, it is moved to position b.  11/13/2018

Voltage on L RL Circuits ( off) Max = - 37% Max at t=L/R Current Max = /R 37% Max at t=L/R I t Voltage on L Max = - 37% Max at t=L/R VL - t 11/13/2018

 (on)  (off) I I t t VL VL t t L/R 2L/R L/R 2L/R /R /R -  t t  VL VL t - t 11/13/2018

Example 2 R I  a b L At t=0 the switch is thrown from position b to position a in the circuit shown: What is the value of the current I a long time after the switch is thrown? R (a) I = 0 (b) I = /2R (c) I = 2/R 11/13/2018

Example 3 R I  a b L At t=0 the switch is thrown from position b to position a in the circuit shown: What is the value of the current I immediately after the switch is thrown? R (a) I = 0 (b) I = /2R (c) I = 2/R 11/13/2018

R  L R  L R  L 3 2 1 2,1,3 1, 2, 3 2, 1, 3 3, 1, 2 1, 3, 2 2, 3, 1 Three circuits with identical batteries, inductors, and resistors: Rank the circuits according to the current through the battery just after the switch is closed, greatest first. 11/13/2018

Inductor in Series L2 i L1 Consider two inductors in series. Both inductors will carry the same current i. 11/13/2018

Inductor in Parallel L1 L2 i i1 i2 Consider two inductors in series. Both inductors are at the same potential. 11/13/2018

Example 4 1 2 (a) t2 < t1 (b) t2 = t1 (c) t2 > t1 R  a b L I 1 2 At t=0 the switch is thrown from b to a as shown: t1 = time for circuit 1 to reach 1/2 of its asymptotic current t2 = time for circuit 2 to reach 1/2 of its asymptotic current. (a) t2 < t1 (b) t2 = t1 (c) t2 > t1 11/13/2018

Energy in an Inductor Start with loop rule: L  a b L How much energy is stored in an inductor when a current is flowing through it? Start with loop rule: Multiply this equation by I: PL, the rate at which energy is being stored in the inductor: Integrate this equation to find U, the energy stored in the inductor when the current = I: 11/13/2018

Where is the Energy Stored? Claim: (without proof) energy is stored in the magnetic field itself (just as in the capacitor / electric field case). To calculate this energy density, consider the uniform field generated by a long solenoid: l r N turns The inductance L is: Energy U: The energy density is found by dividing U by the volume containing the field: 11/13/2018

Self Inductance The magnetic field produced by the current in the loop shown is proportional to that current. I The flux is also proportional to the current. Define the constant of proportionality between flux and current to be the inductance, L We can define the inductance, L, using Faraday's Law, in terms of the emf induced by a changing current. 11/13/2018

Calculation of Inductance (from last class) *Long solenoid with N turns, radius r, length L: l r N turns 11/13/2018

Example 1 Consider the two inductors shown: N turns 2l 2N turns Consider the two inductors shown: Inductor 1 has length l, N total turns and has inductance L1. Inductor 2 has length 2l, 2N total turns and has inductance L2. What is the relation between L1 and L2? (a) L2 < L1 (b) L2 = L1 (c) L2 > L1 11/13/2018

Self Inductance r << l The inductance of an inductor can be calculated from its geometry alone if the device is constructed from conductors and air. If extra material (e.g. iron core) is added, then we need to add some knowledge of materials as we did for capacitors (dielectrics) and resistors (resistivity) Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor d A - - - - - + + + + l r N turns r << l 11/13/2018

Mutual Inductance A changing current in a coil induces an emf in an adjacent coil. The coupling between the coils is described by mutual inductance M. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/indmut.gif M depends only on geometry of the coils (size, shape, number of turns, orientation, separation between the coils).

Magnetic Flux Through a Superconducting Ring Cool it down Move magnet What will happen? Odd consequences due to R=0 The induced current in the loop produces an induced B field.

Superconductors Meissner effect (1939) Equal and opposite B field --- Meissner effect (1939) B field becomes =0 because superconducting currents on the surface of the superconductor create a magnetic field that cancel the applied one for T<Tc 11/13/2018

magnetic levitation of a type 1 superconductor DEMO Magnet: NdFeB Magnetic levitation: Repulsion between the permanent Magnetic field producing the applied field and the magnetic field produced by the currents induced in the superconductor Superconductor: YBa2Cu3O7 11/13/2018