Electromagnetic Induction

emf – electromotive force When you studied electric circuits, you learned that a source of electrical energy, such as a battery is needed to produce a continuous current. When you studied electric circuits, you learned that a source of electrical energy, such as a battery is needed to produce a continuous current. The potential difference, or voltage given to the charges by the battery is called the electromotive force, or emf. The potential difference, or voltage given to the charges by the battery is called the electromotive force, or emf. Emf is not actually a force, it is a potential difference measured in volts. Emf is not actually a force, it is a potential difference measured in volts.

Faraday’s Experiment - Induction A current can be produced by a changing magnetic field (not a constant B field!) A current can be produced by a changing magnetic field (not a constant B field!) First shown in an experiment by Michael Faraday First shown in an experiment by Michael Faraday A primary coil is connected to a battery A primary coil is connected to a battery A secondary coil is connected to an ammeter A secondary coil is connected to an ammeter From Ch. 24, a B field is produced by a moving charge. Well it works the other way too. From Ch. 24, a B field is produced by a moving charge. Well it works the other way too.

Faraday’s Experiment The secondary circuit has a current that is induced by the magnetic field that is created by current through the primary coil. The secondary circuit has a current that is induced by the magnetic field that is created by current through the primary coil.  When the switch is closed, the ammeter deflects in one direction and then returns to zero. It responds to the changing magnetic field as the primary current ramps up.  When there is a steady current in the primary circuit, the ammeter reads zero.  When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero (as the current ramps down.)

Faraday’s Conclusions An electrical current is produced by a changing magnetic field An electrical current is produced by a changing magnetic field The secondary circuit acts as if a source of emf were connected to it for a short time The secondary circuit acts as if a source of emf were connected to it for a short time It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field

Magnetic Flux The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field. The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field. Magnetic flux is defined in a manner similar to that of electrical flux. Magnetic flux is defined in a manner similar to that of electrical flux. Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop. Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop.

Magnetic Flux, 2 The box is a loop of wire. The box is a loop of wire. The wire is in a uniform magnetic field B. The wire is in a uniform magnetic field B. The loop has an area A. The loop has an area A. The flux is defined as The flux is defined as Φ B = B  A = B A cos θ Φ B = B  A = B A cos θ θ is the angle between B and the normal to the plane θ is the angle between B and the normal to the plane Flux ONLY counts the component of the B field that is perpendicular to the loop Flux ONLY counts the component of the B field that is perpendicular to the loop

Magnetic Flux, 3 When the field is perpendicular to the plane of the loop, as in (a), θ = 0 and Φ B = Φ B, max = BA When the field is parallel to the plane of the loop, as in b, θ = 90° and Φ B = 0 The flux can be negative, for example if θ = 180° SI units of flux are T m² = Wb (Weber) (so if you see the unit of Weber/m2, that is a Tesla) Φ B = B  A = B A cos θ θ is the angle between B and the normal to the plane of the loop (one loop).

Magnetic Flux The flux can be visualized with respect to magnetic field lines The flux can be visualized with respect to magnetic field lines The value of the magnetic flux is proportional to the total number of lines passing through the loop The value of the magnetic flux is proportional to the total number of lines passing through the loop When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is parallel to the lines, no lines pass through the area and the flux is 0 When the area is parallel to the lines, no lines pass through the area and the flux is 0

emf and magnetic flux

Electromagnetic Induction – An Experiment When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) When the magnet is held stationary, there is no current (b) When the magnet is held stationary, there is no current (b) When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) If the loop is moved instead of the magnet, a current is also detected If the loop is moved instead of the magnet, a current is also detected

Induction Note the direction of the induced current. Note the direction of the induced current. Top picture, N end magnet moves towards loop. Top picture, N end magnet moves towards loop. Use RHR B on the current in the loop. Your thumb will point in opposite direction of motion of magnet. Use RHR B on the current in the loop. Your thumb will point in opposite direction of motion of magnet. Induced current always opposes change in B field (that’s why there’s a negative sign in the eq.) Induced current always opposes change in B field (that’s why there’s a negative sign in the eq.)

Electromagnetic Induction – Results of the Experiment A current is set up in the circuit as long as there is relative motion between the magnet and the loop A current is set up in the circuit as long as there is relative motion between the magnet and the loop The same experimental results are found whether the loop moves or the magnet moves The same experimental results are found whether the loop moves or the magnet moves The current is called an induced current because is it produced by an induced emf The current is called an induced current because is it produced by an induced emf

PhET simulations on induction faraday_en.jar faraday_en.jar faraday_en.jar  Go to “pickup coil” tab  Note that these are electrons moving in the coil, so use your left hand.

Applications of Faraday’s Law – Ground Fault Interrupters The ground fault interrupter (GFI) is a safety device that protects against electrical shock The ground fault interrupter (GFI) is a safety device that protects against electrical shock Wire 1 leads from the wall outlet to the appliance Wire 1 leads from the wall outlet to the appliance Wire 2 leads from the appliance back to the wall outlet Wire 2 leads from the appliance back to the wall outlet The iron ring confines the magnetic field, which is generally 0 (B fields from currents in opposite directions cancel out) The iron ring confines the magnetic field, which is generally 0 (B fields from currents in opposite directions cancel out) If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current

Applications of Faraday’s Law – Electric Guitar pickup A magnetized vibrating string induces an emf in a coil A magnetized vibrating string induces an emf in a coil A permanent magnet inside the coil magnetizes the portion of the string nearest the coil A permanent magnet inside the coil magnetizes the portion of the string nearest the coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil The changing flux produces an induced emf that is fed to an amplifier The changing flux produces an induced emf that is fed to an amplifier

Applications of Faraday’s Law – Apnea Monitor The coil of wire attached to the chest carries an alternating current The coil of wire attached to the chest carries an alternating current An induced emf produced by the varying field passes through a pick up coil An induced emf produced by the varying field passes through a pick up coil When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert

Application of Faraday’s Law – Motional emf A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field RHR says + charges go up RHR says + charges go up LHR says - charges go down LHR says - charges go down The electrons in the conductor experience a magnetic force The electrons in the conductor experience a magnetic force F = q v B F = q v B The electrons move to the lower end of the conductor (LHR) The electrons move to the lower end of the conductor (LHR) B field goes into page l

Motional emf As the - charges accumulate at the base, that is like + charge moving upwards. As the - charges accumulate at the base, that is like + charge moving upwards. As a result of this charge separation, an electric field is produced in the conductor (see above) As a result of this charge separation, an electric field is produced in the conductor (see above) Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force F=qE = qvB so E = vB Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force F=qE = qvB so E = vB There is a potential difference between the upper and lower ends of the conductor V=E* l (recall that E = Volts/meter, so V = E*length) There is a potential difference between the upper and lower ends of the conductor V=E* l (recall that E = Volts/meter, so V = E*length) E

Motional emf Therefore the potential difference between the ends of the conductor can be found by Therefore the potential difference between the ends of the conductor can be found by emf = ΔV = E l = B l v emf = ΔV = E l = B l v (ΔV = voltage or emf, L= length of wire, v=velocity) (ΔV = voltage or emf, L= length of wire, v=velocity) The upper end is at a higher potential than the lower end The upper end is at a higher potential than the lower end A potential difference is maintained across the conductor as long as there is motion through the field 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 If the motion is reversed, the polarity of the potential difference is also reversed

A wire of length L, moving through a magnetic field, B, induces an electromotive force (EMF=Voltage). If the wire is part of a circuit, then there will be a current in the direction shown (+ charge moves downwards by RHR) This current will interact with the magnetic field and produce a force, F = BIL. Notice that the resulting force F= BIL, opposes the original motion, v, of the wire.

Motional emf in a closed loop circuit EMF

Motional emf in a closed loop circuit ׀ε׀ = EMF = B l v

Lenz’ Law Revisited- opposing the motion As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases The induced current must be in a direction such that it opposes the change in the external magnetic flux The induced current must be in a direction such that it opposes the change in the external magnetic flux This is why there is a negative sign in Faraday’s law This is why there is a negative sign in Faraday’s law Original: Use RHR A: fingers into, thumb right, palm up, therefore current flows up. Induced: Use RHR A: fingers in, thumb up, palm left, induced force goes left.

Lenz’ Law, Opposing the Motion If the magnetic flux due to the external field is increasing into the page, then the flux due to the induced current must be out of the page If the magnetic flux due to the external field is increasing into the page, then the flux due to the induced current must be out of the page Therefore the current must be counterclockwise when the bar moves to the right Therefore the current must be counterclockwise when the bar moves to the right (RHR B: Put thumb in direction of current, curl fingers, they point out of page!)

Lenz’ Law, Opposing the Motion Now imagine the bar is moving toward the left (less area) Now imagine the bar is moving toward the left (less area) The magnetic flux through the loop is decreasing with time The magnetic flux through the loop is decreasing with time The induced current must be clockwise to to produce its own flux into the page The induced current must be clockwise to to produce its own flux into the page RHR B: Thumb in direction of I, curl fingers, and they go into page (within the loop) adding to flux.

Lenz’ Law Revisited, Proof by Counterexample Assume the bar is moving to the right Assume the bar is moving to the right What if the induced current was clockwise? What if the induced current was clockwise? The magnetic force on the bar would be to the right The magnetic force on the bar would be to the right The force would cause an acceleration and the velocity would increase The force would cause an acceleration and the velocity would increase This would cause the flux to increase and the current to increase and the velocity to increase… This would cause the flux to increase and the current to increase and the velocity to increase… This would violate Conservation of Energy and so therefore, the current must be counterclockwise, as shown on the last slides This would violate Conservation of Energy and so therefore, the current must be counterclockwise, as shown on the last slides

Lenz’ Law, Moving Magnet Example A bar magnet is moved to the right toward a stationary loop of wire (a) (the diagram on the left) As the magnet gets closer, magnetic flux increases The induced current (note direction of I) produces a flux to the left, so the current is in the direction shown for loop (b) Note B field for (b) opposes change in B field for (a)

Eddy Current Videos 2’ Neodymium magnet in a copper pipe 2’ Neodymium magnet in a copper pipe ’ Eddy current demo w/ Alum pendulum 4’ Eddy current demo w/ Alum pendulum ’ Russian guy neodymium magnet/Eddy current 2’ Russian guy neodymium magnet/Eddy current

Lenz’ Law, Final Note When applying Lenz’ Law, there are two magnetic fields to consider When applying Lenz’ Law, there are two magnetic fields to consider The external changing magnetic field that induces the current in the first loop The external changing magnetic field that induces the current in the first loop The magnetic field produced by the current in the second loop The magnetic field produced by the current in the second loop

Generators Alternating Current (AC) generator Alternating Current (AC) generator Converts mechanical energy to electrical energy Converts mechanical energy to electrical energy Consists of a wire loop rotated by some external mechanical means Consists of a wire loop rotated by some external mechanical means There are a variety of sources that can supply the energy to rotate the loop There are a variety of sources that can supply the energy to rotate the loop These may include falling water, heat by burning coal to produce steam, nuclear power, wind power, etc. These may include falling water, heat by burning coal to produce steam, nuclear power, wind power, etc.

AC Generators As the loop rotates, the magnetic flux through it changes with time (max when it’s perpendicular to the B field) As the loop rotates, the magnetic flux through it changes with time (max when it’s perpendicular to the B field) This induces an emf, which creates a current in the external circuit that runs the toaster This induces an emf, which creates a current in the external circuit that runs the toaster The ends of the loop are connected to slip rings that rotate with the loop The ends of the loop are connected to slip rings that rotate with the loop Connections to the external circuit are made by stationary brushes in contact with the slip rings Connections to the external circuit are made by stationary brushes in contact with the slip rings

Generators

Generators (cont.)

AC Generators – Summary

DC Generators Components are essentially the same as that of an AC generator Components are essentially the same as that of an AC generator The major difference is the contacts to the rotating loop are made by a split ring or commutator. The major difference is the contacts to the rotating loop are made by a split ring or commutator.

DC Generators The output voltage The output voltage always has the same polarity. The current is a pulsing The current is a pulsingcurrent To produce a steady current, many loops and commutators around the axis of rotation are used To produce a steady current, many loops and commutators around the axis of rotation are used The multiple outputs are superimposed and the output is almost free of fluctuations (not like the graph above) The multiple outputs are superimposed and the output is almost free of fluctuations (not like the graph above)

Motor example / commutator About 46 minutes into video About 46 minutes into video MIT 8.02 Lecture 11 MIT 8.02 Lecture 11

RMS voltage and current

QUICK QUIZ 1 The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

QUICK QUIZ 1 ANSWER (b), (c), (a). At each instant, the magnitude of the induced emf is proportional to the rate of change of the magnetic field (hence, proportional to the slope of the curve shown on the graph).