Magnetism Equations F = qvB sin q E = F/q E = V/d F = ma ac = v2/r For a magnetic force/field F = qvB sin q For an electric field If a particle in a field In a capacitor E = F/q E = V/d Other useful formulas F = ma ac = v2/r

Magnetism Part IV The Final Frontier

Magnetic Flux The product of magnetic field and area. Can be thought of as a total magnetic “effect” on a coil of wire of a given area. B A

Maximum Flux The area is aligned so that a perpendicular to the area points parallel to the field B A

Minimum Flux The area is aligned so that a perpendicular to the area points perpendicular to the field B A

Intermediate Flux The area is neither perpendicular nor is it parallel B A

Magnetic Flux FB = B A cos FB = BA FB: magnetic flux in Webers (Tesla meters2) B: magnetic field in Tesla A: area in meters2. : the angle between the area and the magnetic field. FB = BA

Sample Problem Calculate the magnetic flux through a rectangular wire frame 3.0 m long and 2.0 m wide if the magnetic field through the frame is 4.2 mT. Assume that the magnetic field is perpendicular to the area vector. Assume that the magnetic field is parallel to the area vector. Assume that the angle between the magnetic field and the area vector is 30o.

Sample Problem Assume the angle is 40o, the magnetic field is 50 mT, and the flux is 250 mWb. What is the radius of the loop? B A

Induced Electric Potential A system will respond so as to oppose changes in magnetic flux. A change in magnetic flux will be partially offset by an induced magnetic field whenever possible. Changing the magnetic flux through a wire loop causes current to flow in the loop. This is because changing magnetic flux induces an electric potential.

Faraday’s Law of Induction e = -NDFB/Dt e: induced potential (V) N: # loops FB: magnetic flux (Webers, Wb) t: time (s)

A closer look … e = -DF/Dt e = -D(BAcos)/Dt To generate voltage Change B Change A Change 

Sample Problem A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

Sample Problem A single coil of radius 0.25 m is in a 100 mT magnetic field such that the flux is maximum. At time t = 1.0 seconds, field increases at a uniform rate so that at 11 seconds, it has a value of 600 mT. At time t = 11 seconds, the field stops increasing. What is the induced potential A) at t = 0.5 seconds? B) at t = 3.0 seconds? C) at t = 12 seconds?

Lenz’s Law The current will flow in a direction so as to oppose the change in flux. Use in combination with hand rule to predict current direction.

Sample Problem The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

Sample Problem The magnetic field is increasing at a rate of 4.0 mT/s. What is the direction of the current in the wire loop?

Sample Problem The magnetic field is decreasing at a rate of 4.0 mT/s. The radius of the loop is 3.0 m, and the resistance is 4 W. What is the magnitude and direction of the current?

Work Time Work on Review Packets. Work on Exam Corrections. Do NOT work on tonight’s homework!

Motional emf e = BLv B: magnetic field (T) L: length of bar moving through field v: speed of bar moving through field. Bar must be “cutting through” field lines. It cannot be moving parallel to the field. This formula is easily derivable from Faraday’s Law of Induction

Motional emf - derivation e = DFB/Dt e = D(BA) /Dt (assume cosq = 1) e = D(BLx) /Dt e = BLDx /Dt e = BLv

Sample Problem B = 0.15 T 50 cm 3 W v = 2 m/s Assume the rod is being pulled so that it is traveling at a constant 2 m/s. How much force must be applied to keep it moving at this constant speed?  B = 0.15 T 50 cm 3 W v = 2 m/s

Sample Problem B = 0.15 T 50 cm 3 W v = 2 m/s How much current flows through the resistor? How much power is dissipated by the resistor?  B = 0.15 T 50 cm 3 W v = 2 m/s

Induced emf A current can be produced by a changing magnetic field First shown in an experiment by Michael Faraday A primary coil is connected to a battery A secondary coil is connected to an ammeter

Faraday’s Experiment The purpose of the secondary circuit is to detect current that might be produced by the magnetic field When the switch is closed, the ammeter deflects in one direction and then returns to zero When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Conclusions 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 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 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, 2 You are given a loop of wire The wire is in a uniform magnetic field B The loop has an area A The flux is defined as ΦB = BA = B A cos θ θ is the angle between B and the normal to the plane

Magnetic Flux, 3 When the field is perpendicular to the plane of the loop, as in a, θ = 0 and ΦB = ΦB, max = BA When the field is parallel to the plane of the loop, as in b, θ = 90° and ΦB = 0 The flux can be negative, for example if θ = 180° SI units of flux are T m² = Wb (Weber)

Magnetic Flux, final 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 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

Electromagnetic Induction – An Experiment 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 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

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 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

Faraday’s Law and Electromagnetic Induction The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit If a circuit contains N tightly wound loops and the flux changes by ΔΦ during a time interval Δt, the average emf induced is given by Faraday’s Law:

Faraday’s Law and Lenz’ Law The change in the flux, ΔΦ, can be produced by a change in B, A or θ Since ΦB = B A cos θ The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ 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

Applications of Faraday’s Law – Ground Fault Interrupters 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 2 leads from the appliance back to the wall outlet The iron ring confines the magnetic field, which is generally 0 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 A vibrating string induces an emf in a coil A permanent magnet inside the coil magnetizes a 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 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 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

Application of Faraday’s Law – Motional emf A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field The electrons in the conductor experience a magnetic force F = q v B The electrons tend to move to the lower end of the conductor

QUICK QUIZ 20.1 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 20.1 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).

Lenz’ Law Revisited – Moving Bar Example 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 in a direction such that it opposes the change in the external magnetic flux

Lenz’ Law, Bar Example, cont The flux due to the external field in increasing into the page 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

Lenz’ Law, Bar Example, final The bar is moving toward the left 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

Lenz’ Law Revisited, Conservation of Energy Assume the bar is moving to the right Assume the induced current is clockwise The magnetic force on the bar would be to the right 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 violate Conservation of Energy and so therefore, the current must be counterclockwise

Lenz’ Law, Moving Magnet Example A bar magnet is moved to the right toward a stationary loop of wire (a) As the magnet moves, the magnetic flux increases with time The induced current produces a flux to the left, so the current is in the direction shown (b)

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

Generators Alternating Current (AC) generator Converts mechanical energy to electrical energy Consists of a wire loop rotated by some external means 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

AC Generators, cont Basic operation of the generator As the loop rotates, the magnetic flux through it changes with time This induces an emf and a current in the external circuit The ends of the loop are connected to slip rings that rotate with the loop Connections to the external circuit are made by stationary brushed in contact with the slip rings

AC Generators, final The emf generated by the rotating loop can be found by ε =2 B ℓ v=2 B ℓ sin θ If the loop rotates with a constant angular speed, ω, and N turns ε = N B A ω sin ω t ε = εmax when loop is parallel to the field ε = 0 when when the loop is perpendicular to the field

DC Generators 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

DC Generators, cont The output voltage always has the same polarity The current is a pulsing current 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

Motors Motors are devices that convert electrical energy into mechanical energy A motor is a generator run in reverse A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

Motors and Back emf The phrase back emf is used for an emf that tends to reduce the applied current When a motor is turned on, there is no back emf initially The current is very large because it is limited only by the resistance of the coil

Motors and Back emf, cont As the coil begins to rotate, the induced back emf opposes the applied voltage The current in the coil is reduced The power requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads