Chapter 31 Faradays Law Faradays Experiment A primary

Chapter 31 Faraday’s Law

Faraday’s Experiment • A primary coil is connected to a battery and a secondary coil is connected to an ammeter • The purpose of the secondary circuit is to detect current that might be produced by a (changing) magnetic field • When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Experiment • When the switch is opened, the ammeter reads a current and then returns to zero • When the switch is closed, the ammeter reads a current in the opposite direction and then returns to zero • An induced emf is produced in the secondary circuit by the changing magnetic field

Electromagnetic Induction • When a magnet moves toward a loop of wire, the ammeter shows the presence of a current • When the magnet moves away from the loop, the ammeter shows a current in the opposite direction • When the magnet is held stationary, there is no current • If the loop is moved instead of the magnet, a current is also detected

Electromagnetic Induction • A current is set up in the circuit as long as there is relative motion between the magnet and the loop • The current is called an induced current because is it produced by an induced emf

Faraday’s Law and Electromagnetic Induction • Faraday’s law of induction: the instantaneous emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit • If the circuit consists of N loops, all of the same area, and if FB is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

Faraday’s Law and Lenz’ Law • The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law: • The current caused by the induced emf travels in the direction that creates a magnetic field with flux opposing the change in the original flux through the circuit Heinrich Friedrich Emil Lenz 1804 – 1865

Faraday’s Law and Lenz’ Law • Example: • The magnetic field, B, becomes smaller with time and this reduces the flux • The induced current will produce an induced field, Bind, in the same direction as the original field

Faraday’s Law and Lenz’ Law • Example: • Assume a loop enclosing an area A lies in a uniform magnetic field • Since ΦB = B A cos θ, the change in the flux, ΔΦB, can be produced by a change in B, A or θ

Chapter 31 Problem 17 A coil formed by wrapping 50 turns of wire in the shape of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of 30. 0° with the direction of the field. When the magnetic field is increased uniformly from 200 μT to 600 μT in 0. 400 s, an emf of magnitude 80. 0 m. V is induced in the coil. What is the total length of the wire?

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 FB = q v B • The electrons tend to move to the lower end of the conductor • As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor

Motional emf • 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 FE = q v B; E = v B; • There is a potential difference between the upper and lower ends of the conductor

Motional emf • The potential difference between the ends of the conductor (the upper end is at a higher potential than the lower end): ΔV = E ℓ = B ℓ v • 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

Motional emf in a Circuit • As the bar (with zero resistance) is pulled to the right with a constant velocity under the influence of an applied force, the free charges experience a magnetic force along the length of the bar • This force sets up an induced current because the charges are free to move in the closed path • The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop

Motional emf in a Circuit • The induced, motional emf, acts like a battery in the circuit • 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)

Motional emf in a Circuit • 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 • If the bar is moving toward the left, the magnetic flux through the loop is decreasing with time – the induced current must be clockwise to produce its own flux into the page

Motional emf in a Circuit • The applied force does work on the conducting bar, thus moving the charges through a magnetic field and establishing a current • The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work • The power input is equal to the rate at which energy is delivered to the resistor

Chapter 31 Problem 24 A conducting rod of length ℓ moves on two horizontal frictionless rails. A constant force of magnitude 1. 00 N moves the bar at a uniform speed of 2. 00 m/s through a magnetic field that is directed into the page. (a) What is the current in an 8. 00 -Ω resistor R? (b) What is the rate of energy dissipation in the resistor? (c) What is the mechanical power delivered by the constant force?

Induced emf and Electric Fields • An electric field is created in the conductor as a result of the changing magnetic flux • Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space (this induced electric field is nonconservative, unlike the electric field produced by stationary charges) • The emf for any closed path can be expressed as the line integral • Faraday’s law can be written in a general form:

Lenz’ Law – Moving Magnet Example • As the bar magnet is moved to the right toward a stationary loop of wire, the magnetic flux increases with time • The induced current produces a flux to the left, so the current is in the direction shown • When applying Lenz’ Law, there are two magnetic fields to consider: changing external and induced

Lenz’ Law – Rotating Loop Example • Assume a loop with N turns, all of the same area rotating in a magnetic field • The flux through the loop at any time t is FB = BAcosq = BAcoswt • The induced emf in the loop is • This is sinusoidal, with emax = NABw

AC Generators • Alternating Current (AC) generators convert mechanical energy to electrical energy • Consist of a wire loop rotated by some external means (falling water, heat by burning coal to produce steam, etc. ) • As the loop rotates, the magnetic flux through it changes with time inducing an emf and a current in the external circuit

AC Generators • 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 • The emf generated by the rotating loop:

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 • The output voltage always has the same polarity • The current is a pulsing current

DC Generators • 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 (generators run in reverse) • A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device • As the coil begins to rotate, the induced back emf opposes the applied voltage and the current in the coil is reduced • The induced emf explains why the power requirements for starting a motor and for running it are greater for heavy loads than for light ones

Answers to Even Numbered Problems Chapter 31: Problem 28 (a) to the right (b) out of the plane of the paper (c) to the right (d) into the paper