General Physics (PHY 2140) Lecture 9 Electricity and
General Physics (PHY 2140) Lecture 9 Electricity and Magnetism Induced voltages and induction Magnetic flux and induced emf Faradays law Motional EMF Lenzs Law Applications Self-Inductance RL Circuits (maybe) http://www.physics.wayne.edu/~alan/2140Website/Main.htm Chapter 20 02/04/20 1 Lightning Review Last lecture: 1. Magnetism Application of magnetic forces Amperes law
Bl o I Current loops and solenoids B A BA cos 2. Induced voltages and induction Magnetic flux Review Problem: A rectangular loop is placed in a uniform magnetic field with the plane of the loop parallel to the direction of the field. If a current is made to flow through the loop in the sense shown by the arrows, the field exerts on the loop: 1. a net force. 2. a net torque. 3. a net force and a net torque. 4. neither a net force nor a net torque. 02/04/20 Recall: F qvB sin
2 Magnetic Field of a current loop Magnetic field produced by a wire can be enhanced by having the wire in a loop. x1 B I x2 02/04/20 3 Solenoid Magnet Field lines inside a solenoid magnet are parallel, uniformly spaced and close together. The field inside is uniform and strong. The field outside is non uniform and much weaker.
One end of the solenoid acts as a north pole, the other as a south pole. For a long and tightly looped solenoid, the field inside has a value: B o nI 02/04/20 4 Solenoid Magnet B o nI n = N/l : number of (loop) turns per unit length. I : current in the solenoid. 7 o 4 10 Tm / A 02/04/20 5 Chapter 20: Introduction
Previous chapter: electric currents produce magnetic fields (Oersteds experiments) Is the opposite true: can magnetic fields create electric currents? 02/04/20 6 20.1 Induced EMF and magnetic flux Definition of Magnetic Flux Just like in the case of electric flux, consider a situation where the magnetic field is uniform in magnitude and direction. Place a loop in the B-field. The flux, , is defined as the product of the field magnitude by the area crossed by the field lines. B perp A BA cos where B perp is the component of B perpendicular to the loop, is the angle
between B and the normal to the loop. Units: Tm2 or Webers (Wb) 02/04/20 The value of magnetic flux is proportional to the total number of magnetic field lines passing through the loop. 7 Problem: determining a flux A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0 with the normal to the plane of the loop, determine the magnetic flux through the loop. 02/04/20 8 A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0 with the normal to the plane of the loop, determine the magnetic flux through the loop.
Solution: Given: From what we are given, we use L = 2.00 m B = 0.300 T = 50.0 BA cos 0.300T 2.00m cos 50.0 2 0.386 Tm 2 Find: =? 02/04/20 9 20.1 Induced EMF and magnetic flux
Faradays experiment Picture Molecular Expressions 02/04/20 Two circuits are not connected: no current? However, closing the switch we see that the compass needle moves and then goes back to its previous position Nothing happens when the current in the primary coil is steady But same thing happens when the switch is opened, except for the needle going in the opposite direction What is going on? 10
20.2 Faradays law of induction Induced current I 02/04/20 v S N 11 20.2 Faradays law of induction I v B S
I N B I 02/04/20 v A current is set up in the circuit as long as there is relative motion between the magnet and the loop. 12 Does there have to be motion? (induced) I I
- + AC Delco 1 volt 02/04/20 13 Does there have to be motion? I - + AC Delco 1 volt
02/04/20 14 Does there have to be motion? I (induced) - + AC Delco 1 volt 02/04/20 15 Does there have to be motion?
NO!! - + AC Delco 1 volt 02/04/20 16 Maybe the B-field needs to change.. B 02/04/20 v 17 Maybe the B-field needs to change..
I B 02/04/20 v 18 Maybe the B-field needs to change.. I I v B 02/04/20 I 19
Faradays law of magnetic induction In all of those experiment induced EMF is caused by a change in the number of field lines through a loop. In other words, The instantaneous EMF induced in a circuit equals the rate of change of magnetic flux through the circuit. E Lenzs law N t The number of loops matters Lenzs Law: The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the circuit. 02/04/20 20
Applications: Ground fault interrupter Electric guitar SIDS monitor Metal detector 02/04/20 21 Example : EMF in a loop A wire loop of radius 0.30m lies so that an external magnetic field of strength +0.30T is perpendicular to the loop. The field changes to -0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through the loop.) Find the magnitude of the average induced emf in the loop during this time. 02/04/20
B 22 A wire loop of radius 0.30m lies so that an external magnetic field of strength +0.30T is perpendicular to the loop. The field changes to -0.20T in 1.5s. (The plus and minus signs here refer to opposite directions through the loop.) Find the magnitude of the average induced emf in the loop during this time. Given: r = 0.30 m Bi = 0.30 T Bf = -0.20 T t = 1.5 s The loop is always perpendicular to the field, so the normal to the loop is parallel to the field, thus cos = 1. The flux is then BA B r 2 Initially the flux is 2
i 0.30T 0.30m = 0.085 T m 2 Find: EMF=? and after the field changes the flux is 2 f 0.20T 0.30m = -0.057 T m 2 The magnitude of the average induced emf is: f i 0.085 T m 2 -(-0.057 T m 2 ) emf 0.095V t t 1.5s 02/04/20 23
Example 2: EMF of a flexible loop The flexible loop in figure below has a radius of 12cm and is in a magnetic field of strength 0.15T. The loop is grasped at points A and B and stretched until it closes. If it takes 0.20s to close the loop, find the magnitude of the average induced emf in it during this time. X X X A X X X X X
X X B X X X X X X = BA = 0.15T (0.12 m)2 = 0.0068 Tm2 t = 0.20 s Then, emf = / t = 0.034 V emf = 0.034 V 02/04/20
24 20.3 Motional EMF l B F v Let's consider a conducting bar moving perpendicular to a uniform magnetic field with constant velocity v. F qvB sin 02/04/20 This force will act on free charges in the conductor. It will tend to move negative charge to one end, and leave the other end of the bar
with a net positive charge. 25 Motional EMF The separated charges will create an electric field which will tend to pull the charges back together. When equilibrium exists, the magnetic force, F=qvB, will balance the electric force, F=qE, such that a free charge in the bar will feel no net force. (recall our velocity selector example) Thus, at equilibrium, E = vB. The potential difference across the ends of the bar is given by V=El or V El Blv A potential difference is maintained across the conductor as long as there is motion through the field. If the motion is reversed, the polarity of the potential difference is also reversed. 02/04/20 26
Motional EMF conducting rails R x B v We can apply Faraday's law to the complete loop. The change of flux through the loop is proportional to the change of area from the motion of the bar: BA Bl x 02/04/20 current I or (Faradays law)
E Blv R R x E Bl Blv t t Motional EMF 27 Example: wire in the magnetic field Over a region where the vertical component of the Earth's magnetic field is 40.0T directed downward, a 5.00 m length of wire is held in an east-west direction and moved horizontally to the north with a speed of 10.0 m/s. Calculate the potential difference between the ends of the wire, and
determine which end is positive. EMF = Blv = 40.0 T 5.00 m 10.0 m/s = 0.002 V Use right hand rule and F = qv B to determine the polarity. Answer: west end is positive. 02/04/20 28 20.4 Lenzs law revisited Application of Lenz's law will tell us the direction of induced currents, the direction of applied or produced forces, and the polarity of induced emf's. Lenz's law says that the induced current will produce magnetic flux opposing this change. To oppose an increase into the page, it generates magnetic field which points out of the page, at least in the interior of the loop. Such a magnetic field is produced by a counterclockwise current (use the right hand rule to verify).
02/04/20 29 Lenzs law: energy conservation We arrive at the same conclusion from energy conservation point of view The preceding analysis found that the current is moving ccw. Suppose that this is not so. If the current I is cw, the direction of the magnetic force, BlI, on the sliding bar 02/04/20 would be right.
This would accelerate the bar to the right, increasing the area of the loop even more. This would produce even greater force and so on. In effect, this would generate energy out of nothing violating the law of conservation of energy. Our original assertion that the current is cw is not right, so the current is ccw! 30 S S v The induced flux seeks to counteract
the change. N induced change 02/04/20 S S N v N induced change 31 Example: direction of the current Find the direction of the current induced in the resistor at the instant the switch is
closed. Induced current B 02/04/20 32 Applications of Magnetic Induction Tape / Hard Drive / ZIP Readout Tiny coil responds to change in flux as the magnetic domains (encoding 0s or 1s) go by. Question: How can your VCR display an image while paused? Credit Card Reader
Must swipe card generates changing flux Faster swipe bigger signal 02/04/20 33 20.5 Generators Generators and motors are two of the most important applications of induced emf (magnetic inductance). A generator is something that converts mechanical energy to electrical energy. Alternating Current (AC) generator Direct Current (DC) generator A motor does the opposite, it converts electrical energy to mechanical energy.
02/04/20 34 AC generator D C l a Compute EMF It is only generated in BC and DA wires EMF generated in BC and DA would be
EBC EDA Blv perp A B v sin B v Thus, total EMF is E 2 Blv perp 2 Blv sin If the loop is rotating with a E 2 Blv sin t 2 Bl sin t
2 02/04/20 A as v=r=a/ 2 and = t 35 AC generator (cont) Generalize the result to N loops E NBA sin t EMF generated by the AC generator where we also noticed that A=la Emax NBA is Note:
reached when t=90 or 270 (loop parallel to the magnetic field) 02/04/20 36 DC generator By a clever change to the rings and brushes of the ac generator, we can create a dc generator, that is, a generator where the polarity of the emf is always positive. The basic idea is to use a single split ring instead of two complete rings. The split ring is arranged so that, just as the emf is about to change sign from positive to negative, the brushes cross the gap, and the polarity of the contacts is switched. The polarity of the contacts changes in phase with the polarity of the emf -the two changes essentially cancel each other out, and the emf remains always positive. The emf still varies sinusoidally during each half cycle, but every half cycle is a positive emf. 02/04/20
37 DC Generator 02/04/20 38 Motors A motor is basically a generator running in reverse. A current is passed through the coil, producing a torque and causing the coil to rotate in the magnetic field. Once turning, the coil of the motor generates a back emf, just as does the coil of a generator. The back emf cancels some of the applied emf, and limits the current through the coil. 02/04/20 39 Example: coil in magnetic field
A coil of area 0.10 m is rotating at 60 rev/s with its axis of rotation perpendicular to a 0.20T magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the coil with respect to the magnetic field? E NBA sin t Recall: 2 f 1000(0.20)(0.10)(260) = 7540 V Max when loop parallel to the magnetic field. 02/04/20 40 Eddy currents (application) Magnetic Levitation (Maglev) Trains Induced surface (eddy) currents produce field in opposite direction Repels magnet
Levitates train S N eddy current rails Maglev trains today can travel up to 310 mph Twice the speed of Amtraks fastest conventional train! May eventually use superconducting loops to produce B-field No power dissipation in resistance of wires! 02/04/20 41 20.6 Self-inductance When a current flows through a loop, the magnetic field
created by that current has a magnetic flux through the area of the loop. If the current changes, the magnetic field changes, and so the flux changes giving rise to an induced emf. This phenomenon is called self-induction because it is the loop's own current, and not an external one, that gives rise to the induced emf. Faradays law states E N t 02/04/20 42 The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuit Thus, the self-induced EMF must be proportional to the time rate of change of the current I
E L t where L is called the inductance of the device Units: SI: henry (H) If flux is initially zero, 02/04/20 1 H 1 V s A N L N I I 43 Example: solenoid A solenoid of radius 2.5cm has 400 turns and a length of 20 cm.
Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75mV. N B 0 nI 0 I l N B N2A L 0 I l E L 02/04/20 I I E
t t L N B BA 0 IA l = (4x 10-7)(160000)(2.0 x 10-3)/(0.2) = 2 mH = -(75 x 10-3)/ (2.0 x 10-3) = 37.5 A/s 44 Inductor in a Circuit Inductance can be interpreted as a measure of opposition to the rate of change in the current Remember resistance R is a measure of opposition to the current As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current
Therefore, the current doesnt change from 0 to its maximum instantaneously Maximum current: I max 02/04/20 E R 45 20.7 RL Circuits Recall Ohms Law to find the voltage drop on R V IR
(voltage across a resistor) We have something similar with inductors I EL L t (voltage across an inductor) Similar to the case of the capacitor, we get an equation for the current as a function of time (series circuit). E Rt / L I 1 e R 02/04/20 L
R 46 RL Circuit (continued) V Rt / L I 1 e R 02/04/20 47 20.8 Energy stored in a magnetic field The battery in any circuit that contains a coil has to do work to produce a current Similar to the capacitor, any coil (or inductor) would store
potential energy 1 2 PEL LI 2 Summary of the properties of circuit elements. 02/04/20 Resistor Capacitor Inductor units ohm, = V / A farad, F = C / V
henry, H = V s / A symbol R C L relation V=IR Q=CV emf = -L (I / t) power dissipated P = I V = I R = V / R
0 0 energy stored 0 PEC = C V / 2 PEL = L I / 2 48 Example: stored energy A 24V battery is connected in series with a resistor and an inductor, where R = 8.0W and L = 4.0H. Find the energy stored in the inductor (a) when the current reaches its maximum value and (b) one time constant after the switch is closed. 02/04/20
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