Chapter 4 – Time Varying Field and Maxwells Equations Ali Othman Universiti Teknologi MARA Pulau Pinang HP No: 019-4260206 BKBA 4.7(04-382 2507)/BP 7.15 (04-382 2827)

Objectives To learn and understand the concept of electromotive force based on Faraday’s experiments. Faraday is often regarded as the greatest experimental scientist of the 1800s. His contributions to the study of electricity include the invention of the electric motor, generator, and transformer. (Image : MICHAEL FARADAY)

Faraday’s and Lenz’s Laws Faraday’s Law: Changing magnetic field produces electric field (and thus voltage and current). The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit. Lenz’ Law: A changing magnetic field produces another magnetic field that opposes the change.

Faraday’s experiment Magnetic field passing through the loop - there is the current. Static magnetic field – no change in current. 3 ways to change the current in the loop. Use the permanent magnet – move it. Magnetic Induction – switching DC magnetic field, but change the loop

Faraday’s experiment The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit

Faraday’s experiment The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit

Induced EMF (magnetic Induction) A current can be produced by a changing magnetic field [B=f (t)], i.e., B varies over time First shown in an experiment by Michael Faraday A primary coil is connected to a battery. A secondary coil is connected to an ammeter.

Force on or I in the filed A whole picture helps Charge q as source Current I as source Gauss’s Law Ampere’s Law Faraday’s Law Electric field E Magnetic field B Ampere-Maxwell Law Force on q in the field Force on or I in the filed Summarized in Lorentz force Maxwell equations

Overview of Faraday’s Law

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 A static magnetic field produces no current flow (no change in current). A time-varying field produces an induced voltage (electromotive force-emf) in a closed circuit, which causes a flow of current. An electrical current is produced by a changing magnetic field (A current can be produced by a changing magnetic field [B=f (t)], i.e., B varies over time) It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field. Magnetic field passing through the loop -there is the current.

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 wire loop wire and the area of the loop

Magnetic Flux You are given a loop of wire The wire is in an 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 (a) When the field is perpendicular to the plane of the loop, θ = 0 and ΦB = ΦB, max = BA (b) When the field is parallel to the plane of the loop, 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 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 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 it is 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 through each loop changes by dΨ during an interval dt, the average emf induced is given by Faraday’s Law:

Faraday’s Law and Lenz’ Law The minus sign is included because of the polarity of the emf. The induced emf in the coil gives rise to a current whose magnetic field OPPOSES ( Lenz’s law) the change in magnetic flux that produced it

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

Applications 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

Applications of Faraday’s Law – Tape Recorder A magnetic tape moves past a recording and playback head The tape is a plastic ribbon coated with iron oxide or chromium oxide

Lenz’ Law – Moving Bar Example (Right) 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

Lenz’ Law – Moving Bar Example (Right) The flux due to the external field is 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 – Moving Bar Example (Left) 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 - 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) A bar magnet is moved to the right toward a stationary loop of wire. As the magnet moves, the magnetic flux increases with time (b) The induced current produces a flux to the left to counteract the increasing external flux to the right

Lenz’ Law 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

Type of EMF 1) Stationary Loop in Time-varying magnetic field B (Transformer emf) 2) Moving Loop in Static B (motional emf) 3) Moving Loop in Time Varying B

Motional EMF As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor As a result of this charge separation, an electric field is produced in the conductor Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force There is a potential difference between the upper and lower ends of the conductor

Vemf =VBℓ voltage across the conductor Motional emf, cont. Vemf =VBℓ voltage across the conductor   If the motion is reversed, the polarity of the potential difference is also reversed

Magnitude of the Motional emf

Motional emf in a Circuit A conducting bar sliding with v along two conducting rails under the action of an applied force Fapp. The magnetic force Fm opposes the motion, and a counterclockwise current is induced.

Motional emf in a Circuit V The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop The induced, motional emf, acts like a battery in the circuit

Motional Emf To apply motional emf eq. is not always easy; some care must be exercised. The following points should be noted: 1:The integral is zero along the portion of the loop where v=0. Thus dl is taken along the portion of the loop that is cutting the field (along the rod) where v has nonzero value. 2:The direction of the induced current is the same as that of (v x B). The limits of the integral are selected in the opposite direction to the induced current thereby satisfying lenz’s law.

Example – Operating a light bulb Rod and rail have negligible resistance but the bulb has a resistance of 96 , B = 0.80 T, v = 5.0 m/s and ℓ = 1.6 m. Calculate (a) emf in the rod, (b) induced current (c) power delivered to the bulb and (d) the energy used by the bulb in 60 s. (a) Vemf = vBℓ = (5.0 m/s)(0.80 T)(1.6 m)=6.4 V (b) I = Vemf /R = (6.4V)/(96 )=0.067 A (c) P = Vemf /I = (6.4 V)(0.067 A)=0.43 W (d) E = Pt = (0.43 W)(60 s)= 26 J (= 26 Ws)

Exercise Figure shows a rectangular coil moving to the right at speed, v = 2.5 Uy m/s. The left side cuts the flux at 90 degrees, where B1 = 0.3 Uz T, while the right side cuts equal amount of flux in the opposite direction. Classify which type of electromotive force voltage, Vemf is generated in this problem. Find the instantaneous current in the coil and determine its direction.

Exercise Figure shows a rectangular coil moving to the right at speed, v = 2.5 Uy m/s. The left side cuts the flux at 90 degrees, where B1 = 0.3 Uz T, while the right side cuts equal amount of flux in the opposite direction. Classify which type of electromotive force voltage, Vemf is generated in this problem. (MOVING loop in static B) Find the instantaneous current in the coil and determine its direction.

Exercise Find the instantaneous current in the coil and determine its direction.

Exercise Figure shows an inductor with N turns of radius a is placed in the x-y plane and connected to resistor R. A time-varying magnetic field is applied over the cross sectional surface of inductor. If the magnetic field is given by Determine : The magnetic flux linking a single turn of the inductor, The transformer emf, given that N=20, Bo=0.2Tesla, a=20 and =108 rad/s and The induced current in the circuit for R=10k (assume the wire resistance to be negligibly small)

End of Chapter Visit https://sites.google.com/site/ecoursef