Lecture 3-5 Faraday’ s Law (pg. 24 – 35) ECT1026 Field Theory Lecture 3-5 Faraday’ s Law (pg. 24 – 35) In 1831, Michael Faraday discovers that a changing magnetic flux can induce an electromotive force.

? Magnetic Field  Electric Current In the previous lectures ECT1026 Field Theory 2007/2008 In the previous lectures Electric Current  Magnetic Field How to determine the magnetic field? Biot-Savart Law Ampere’s Law Long Straight Wire Pie-shaped Wire Loop Circular Loop Wire Long Straight Wire Long Solenoid Toroid Magnetic Field  Electric Current ?

This mechanism is called “Electromagnetic Induction” ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Magnetic Field can produce an electric current in a closed loop, if the magnetic flux linking the surface area of the loop changes with time. This mechanism is called “Electromagnetic Induction” The electric Current Produced  Induced Current

Sensitive current meter ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law First Experiments Move a bar magnet toward the loop, a current suddenly appears in the circuit Conducting loop The current disappears when the bar magnet stops Sensitive current meter If we then move the bar magnet away, a current again suddenly appears, but now in the opposite direction Since there is no battery or other source of emf included, there is no current in the circuit

Discovering of the First Experiments ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Discovering of the First Experiments A current appears only if there is relative motion between the loop and the magnet 2. Faster motion produces a greater current 3. If moving the magnet’s N-pole towards the loop causes clockwise current, then moving the N-pole away causes counterclockwise.

Current in the coil produces a magnetic field B ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law An Experiment - Situation A Current in the coil produces a magnetic field B Constant flux though the loop DC current I, in coil produces a constant magnetic field, in turn produces a constant flux though the loop Constant flux, no current is induced in the loop. No current detected by Galvanometer

Magnetic field drops to zero ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law An Experiment - Situation B: Disconnect battery suddenly Magnetic field drops to zero Deflection of Galvanometer needle Sudden change of magnetic flux to zero causes a momentarily deflection of Galvanometer needle.

An Experiment - Situation C: Reconnect Battery ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law An Experiment - Situation C: Reconnect Battery Current in the coil produces a magnetic field B Sudden change of magnetic flux through the loop Magnetic field becomes non-zero Deflection of Galvanometer needle in the opposite direction Link: http://micro.magnet.fsu.edu/electromag/java/faraday/index.html

Conclusions from the experiment ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Conclusions from the experiment Current induced in the closed loop when magnetic flux changes, and direction of current depends on whether flux is increasing or decreasing If the loop is turned or moved closer or away from the coil, the physical movement changes the magnetic flux linking its surface, produces a current in the loop, even though B has not changed In Technical Terms Time-varying magnetic field produces an electromotive force (emf) which establish a current in the closed circuit

Electromotive force (emf) can be obtained through the following ways: ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Electromotive force (emf) can be obtained through the following ways: 1. A time-varying flux linking a stationary closed path. (i.e. Transformer) 2. Relative motion between a steady flux and a close path. (i.e. D.C. Generator) 3. A combination of the two above, both flux changing and conductor moving simultaneously. A closed path may consists of a conductor, a capacitor or an imaginary line in space, etc.

ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Faraday summarized this electromagnetic phenomenon into two laws ,which are called the Faraday’s law Faraday’s First Law When the flux magnet linked to a circuit changes, an electromotive force (emf) will be induced.

ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Faraday’s Second Law The magnetic of emf induced is equal to the time rate of change of the linked magnetic flux F. (volts) Minus Sign  Lenz’s Law Indicates that the emf induced is in such a direction as to produces a current whose flux, if added to the original flux, would reduce the magnitude of the emf

The induced voltage acts to produce an opposing flux ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Minus Sign  Lenz’s Law The induced voltage acts to produce an opposing flux

The induced voltage acts to produce an opposing flux ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Minus Sign  Lenz’s Law The induced voltage acts to produce an opposing flux

The induced voltage acts to produce an opposing flux ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Minus Sign  Lenz’s Law The induced voltage acts to produce an opposing flux

If the closed path is taken by an N-turn filamentary conductors ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law If the closed path is taken by an N-turn filamentary conductors Magnetic flux ? The magnetic flux  linking a surface S is defined as the total magnetic flux density B passing through S: (Wb) 

ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law From Chapter 2 Electrostatics – Part B (pg 4-5) For a closed loop with contour C, the emf is defined by: Take N = 1  In Electrostatics – an electric field intensity E due to static charge distribution must lead to zero potential difference about a closed path. Here – the line integral leads to a potential difference with time-varying magnetic fields, the results is an emf or a voltage

Faraday’s Law (B and E fields) ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Faraday’s Law (B and E fields) Stationary Loop in a Time-Varying Magnetic Field Moving Conductor in a Static Magnetic Field

Since the loop is stationary, d/dt operates on B(t) only ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Stationary Loop in a Time-Varying Magnetic Field A single-turn (N =1), conducting loop is placed in a time-varying magnetic field B(t). Since the loop is stationary, d/dt operates on B(t) only  Applying Stoke’s theorem to the closed line integral  

Maxwell’s Eqn of Electrostatic ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law If B is not time-varying, i.e. OR Maxwell’s Eqn of Electrostatic

Moving Conductor in a Static Magnetic Field ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Moving Conductor in a Static Magnetic Field A wire with length l moving across a static magnetic field at a constant velocity u (points to x). The conducting wire contains free electron. Magnetic force Fm acting on any charged particle “q” moving with velocity u is:

Em is in a direction perpendicular to the plane containing u and B ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law This Fm is equivalent to the electrical force that would be exerted o the particle by an electric field Em given by: Em is in a direction perpendicular to the plane containing u and B The electric field Em generated by the motion of the charged particle is called a motional electric field

For the wire, Em is along -y ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law For the wire, Em is along -y ^ Magnetic force acting on the electrons in the wire causes them to move in the direction of -Em i.e. towards the end labeled 1 Induces a voltage difference between ends 1 and 2  Voltage induced: motional emf, End 2 being at the higher potential

For the conducting wire: ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law For the conducting wire:  

ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law In general, if any segment of a closed circuit with contour C moves with a velocity u across a static magnetic field B, then the induced motional emf is: Only those segments of the circuit that cross magnetic field lines contribute to

Fleming Right Hand Rule ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Fleming Right Hand Rule Direction of Induced e.m.f, Magnetic Flux, Conductor Motion Fore Finger Direction of Field Flux Thumb Direction of Conductor Motion Middle Finger Direction of Induced emf or Current Flow

Fleming's right hand rule (for generators) Fleming's right hand rule shows the direction of induced current flow when a conductor moves in a magnetic field. The right hand is held with the thumb, first finger and second finger mutually at right angles, as shown in the diagram The Thumb represents the direction of Motion of the conductor. The First finger represents the direction of the Field. The Second finger represents the direction of the induced or generated Current (in the classical direction, from positive to negative).

Fleming's left hand rule (for electric motors) Fleming's left hand rule shows the direction of the thrust on a conductor carrying a current in a magnetic field. The left hand is held with the thumb, index finger and middle finger mutually at right angles. The First finger represents the direction of the Field. The Second finger represents the direction of the Current (in the classical direction, from positive to negative). The Thumb represents the direction of the Thrust or resultant Motion.

Application of Faraday’s Law ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Application of Faraday’s Law Example 3.5-1: The rectangular loop shown in the figure is situated in the x-y plane and moves away from the origin at a velocity (m/s) in a magnetic field given by: (T) If R = 5 , find the current I at the instant that the loop sides are at y1 = 2m and y2= 2.5m . The loop resistance may be ignored.

3.5 Faraday’s Law Example 3.5-1: The induced voltage V12 is given by: ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Example 3.5-1: The induced voltage V12 is given by: Since is along Voltage are induced across only the sides oriented along i.e. sides (1-2) and (3-4) B decreases exponentially with y

3.5 Faraday’s Law Example 3.5-1: Induced voltage V12 is: ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Example 3.5-1: The induced voltage V12 is given by: B decreases exponentially with y At y1 = 2 m Induced voltage V12 is:

3.5 Faraday’s Law Example 3.5-1: Induced voltage V43 is: At y2 = 2.5 m ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Example 3.5-1: At y2 = 2.5 m Induced voltage V43 is: Current in the circuit is:

Example 3.5-2: AC Generator ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Example 3.5-2: AC Generator The Faraday’s Law is the principle at work in an electric generator. The essential design is a conducting coil rotating in the magnetic field of a fixed magnet.

Example 3.5-2: AC Generator ECT1026 Field Theory 2007/2008 3.5 Faraday’s Law Example 3.5-2: AC Generator For constant angular velocity, the magnetic flux through the coil area A is: Conducting Coil B 