# Magnetic Induction Faraday, Lenz, Mutual & Self Inductance Maxwell s Eqns, E-M waves. Reading Journals for Tuesday from table(s)

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1 PHYS Week 12 Magnetic Induction Faraday, Lenz, Mutual & Self Inductance Maxwell s Eqns, E-M waves Reading Journals for Tuesday from table(s) WebAssign due Friday night For exclusive use in PHYS Not for re-distribution. Some materials Copyright University of Colorado, Cengage,, Pearson J. Maps.

2 Magnetic Induction Faraday s, Lenz s Law: An EMF is induced in a loop (or coil) when the number of magnetic field lines passing through it is changing. Magnetic Flux NΦ B = Flux Linkage Faraday s Law (the minus sign is Lenz s Law, N = number of turns for a coil with multiple turns Lenz s Law: The induced current flows to oppose the change in magnetic flux that creates it. (Think first about what induced B needs to be, then induced current) TWO B fields to keep in mind: (1) The external B field (source) (2) The induced B field created by the induced current in loop or coil The direction of the induced current is such that the magnetic field it creates (the induced B field) tries to preserve the magnetic flux at the level from from the initial external B field.

3 Two loop of wires (A and B) are placed near each other. An increasing current in A is turned on. This causes an induced current in loop B, which causes 1. The two loops to repel 2. The two loops to attract 3. Depends on whether the current in A is CW or CCW 4. No net force on either loop B A Adapted from University of Colorado Boulder 0% 0% 0% 0%

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5 A uniform magnetic field is present at left. The U-shaped piece is a conductor fixed in place,.. As the conducting cross bar in (1) is moved upward at constant velocity., an induced current in (1) flows counter-clockwise. What is the direction of the external uniform B field? [Ignore diagram (2) for now.] 1. Into the page 2. Out of the page 3. Parallel to the velocity 4. Opposite to the velocity Adapted from University of Colorado Boulder 0% 0% 0% 0%

6 The two circuits are in the same uniform B field. The width of the U-shaped piece in (1) is 2L. The width of the U is L in (2). The cross bar slides at the same speed in both circuits. Which circuit has the larger induced EMF? 1. (1) 2. (2) 3. The EMFs are equal 4. Depends on resistance of the bars E d dt B Adapted from University of Colorado Boulder 0% 0% 0% 0%

7 Motional EMF/ Slide Generators Mobile charges (electrons) in metal rod deflect down (-) in moving rod due to magnetic force. Slight charge separation creates an electric field that produces a balancing force (Recall Hall Effect) E = vb and a potential difference = vbl When the circuit is completed (below, this serves as an EMF E to drive a current I around a complete circuit: (Variant of Faraday s Law)

8 worksheet

9 A conducting loop is stationary halfway into a magnetic field. Suppose the B-field then begins to increase rapidly in strength. What happens to the loop? 1. It is pushed 2. It is pushed 3. It is pushed 4. It is pushed 5. Nothing no net force Adapted from University of Colorado Boulder 0% 0% 0% 0% 0%

10 Electric fields created by static electric charges are conservative Induced electric fields created by changing magnetic fields are nonconservative. The work done by the induced electric field pushing a charge around a closed path is not zero! Both kinds of electric fields produce a force on a charge: F qe

11 Induced Electric Fields The EMF that creates induced currents is a consequence of electric fields created or induced by the changing magnetic flux. E E E ind E ind dl dl E d dt d dt B B B into page, increasing

12 Mutual Inductance from Faraday s Law E E 2tot 2 1tot 1 M M M M I 1 di dt I 2 di 1 Mutual Inductance, like capacitance, is a geometric factor depending on the spatial arrangement of the coils. dt 2

13 A coil with N c =120 turns, radius 3.0 cm and resistance 5.0 ohms is coaxial with a solenoid with n = 2500 turns/m and radius of 2.0 cm. The current through the solenoid is I s. What is the mutual inductance of this system?

14 Mutual Inductance - Transformers

15 SELF INDUCTANCE MAGNETIC FLUX (LINKAGE) THROUGH A DEVICE DUE TO B FIELD FROM ITS OWN CURRENT Total B LI Ideal self-inductance (inductor) has no electrical resistance UNIT of SELF INDUCTANCE = henry (H) 2 N 2 N 2 N B N( 0 I) R 0 R I

16 Worksheet Find the self-inductance of the solenoid.

17 PHYS Week 12 Magnetic Induction Faraday, Lenz, Mutual & Self Inductance Maxwell s Eqns, E-M waves Reading Journals for Thursday from table(s) WebAssign due Friday night For exclusive use in PHYS Not for re-distribution. Some materials Copyright University of Colorado, Cengage,, Pearson J. Maps.

18 Faraday s Law d2tot 2 MI M tot 1 E2 dt di dt 1 Mutual Inductance of two adjacent coils or circuits (e.g. transformers) E L tot L LI di dt L U I B tot 1 2 LI NBA I 2 u B Self-inductance defined geometric factor like capacitance Φ tot = Total Magnetic Flux through device (linked) due to a device s own current 2 B 2 Faraday s Law Stored potential energy energy density o With voltage drops : V R = IR V C = q / C and V L = L di / dt Then we can still write Σ ΔV = 0 to analyze circuits.

19 Ideal self-inductance (inductor) has no electrical resistance The electric current through an inductor cannot change discontinuously Most real inductors have resistance due to the wire they are wound from, which must be taken into account. (R-L circuits) Large electromagnets wound from superconducting wire CAN behave like ideal inductors. (NMR and MRI systems use these. L = several henries)

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21 Consider two (self) inductors. Inductor 1 consists of a single loop of wire. Inductor 2 is identical to 1, except it has two loops on top of each other. The current is the same in both circuits. What is the B field at the center of coil 2, B 2, compared to the field in the center of coil 1? 1. B 2 = B 1 2. B 2 = 2 B 1 3. B 2 = 4 B 1 4. B 2 = B 1 /2 I Coil 1 Coil 2 I Adapted from University of Colorado Boulder 0% 0% 0% 0%

22 Inductor 1 consists of a single loop of wire. Inductor 2 is identical to 1, except it has two loops on top of each other. The current is the same in both circuits. How do the self-inductances of the two circuits compare? (Recall, L = tot / I = N B A/ I ) 1. L 2 < 2L 1 2. L 2 = 2 L 1 3. L 2 > 2L 1 I Coil 1 Coil 2 I Adapted from University of Colorado Boulder 0% 0% 0%

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24 R-L series circuits. Switch from b to a at t=0. Find I(t) in circuit. I( t) Rt / L I max (1 e ) I max L / Vb R R

25 Which curve represents (1) the voltage reading across the resistor in the RL circuit and (2) the voltage reading across the inductor as functions of time? At t=0 the switch is flipped to position (a) 1. V R = A; V L = B 2. Both V R, V L = A 3. V R = B; V L = A 4. Both V R, V L = B V B A t 0% 0% 0% 0%

26 The switch is closed at t=0. What is the current through the resistor, at t=0+, just an instant after the switch is closed? 1. 0 A A A A A V = 10V R = 20 L = 10H Adapted from University of Colorado Boulder 0% 0% 0% 0% 0%

27 The switch is closed at t=0. What is the initial rate of change of current, di/dt, in the inductor at t = 0+ i.e. just after the switch is closed? (Hint: what is the initial V across the resistor? Inductor?) 1. 0 A/s A/s 3. 1 A/s A/s A/s V = 10V R = 20 L = 10H Adapted from University of Colorado Boulder 0% 0% 0% 0% 0%

28 R-L series circuits. Switch from a to b at t=0 after being at a for a long time. Find I(t) in circuit. I( t) I 0 e Rt / L I 0 Vb R

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30 Biot-Savart and Ampere s Laws disagree about the magnetic field between the plates of a capacitor when it is charging (or discharging). Current I B oi 2r B=? Current I Current I B=? Current I Maxwell proposes: (since changing B can create E) Changing E fields can create B fields Additional term to Ampere s Law, analogous to Faraday s Law

31 Maxwell uses Faraday s Law for insight for an inspired guess: Ampere-Maxwell Law Displacement Current Measured in amperes but not due to real charges in motion Changing Electric Field or Flux INDUCES a Magnetic Field

32 Maxwell s Equations Gauss s Law Ampere-Maxwell Law (Displacement current) Faraday s Law plus

33 Electromagnetic Waves Light and Optics Maxwell s equations predict electromagnetic waves: coupled electric and magnetic fields that travel through space even vacuum -- generated by electric charges that accelerate. The speed of the waves according to Maxwell s equations was predicted to be (in vacuum) 1 8 o o m/s c This corresponded to the speed of light already measured in Maxwell s time And to the speed of soon-to-be discovered radio waves. Wave = some quantity that oscillates with position and time

34 Examples of waves: Sound: oscillating pressure variations coming from oscillating density of molecules in the room. Waves on ropes or strings: wiggle/oscillate one end and the pattern travels along the rope. Stringed instruments exhibit standing waves. Water waves / Ripples on the surface of water: variations in height of the surface that oscillate in time and position The simplest description of waves is for sinusoidal variations in space and time. These can be described by a wave function where y is the quantity that oscillates with position x and time t y( x, t) Asin kxt Asin x t 2 2 T

35 t T x A t kx A t x y 2 2 sin sin ), ( Watch the wave at 1 position (take x = 0, for example) as a function of time: t T A t A t y 2 sin sin ) (0, ), ( ), ( x t x y v t t x y The wave function above is traveling wave and satisfies or is a solution to the wave equation:

36 y( x, t) Take a snapshot of the wave everywhere at a fixed time, say t=0 y( x,0) Asin kxt Asin x t Asin T kx Asin x

37 2 2 y( x, t) Asinkxt Asin x t T Two successive snapshots at t=0 and t=δt (<T) reveal a traveling wave, moving to the right, in +x direction y( x,0) Asin kx y( x, t) Asin kxt

38 The graph below shows a snapshot of a wave on a string that is traveling to the right. There is a bit of string painted at point P. At the instant shown, the velocity of that painted bit of string at P has which direction? y P v wave x Adapted from University of Colorado Boulder 0% 0% 0% 0%

39 Electromagnetic Waves are traveling electric and magnetic fields. Transverse Wave: E and B perpendicular to direction of travel With direction of travel given by: E Bm c 2 k c f m 2f 2 T

40 Electromagnetic Wave Spectrum

41 The electric and magnetic fields of an EM wave are shown at one instant in time. This wave is moving 1. down 2. left 3. toward you 4. away from you 5. Not enough info Adapted from University of Colorado Boulder 0% 0% 0% 0% 0%

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