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Chapter 31 Faraday’s Law

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**Michael Faraday Great experimental physicist and chemist 1791 – 1867**

Contributions to early electricity include: Invention of motor, generator, and transformer Electromagnetic induction Laws of electrolysis: A method of separating bonded elements and compounds by passing an electric current through them

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**Induction An induced current is produced by a changing magnetic field**

There is an induced emf associated with the induced current A current can be produced without a battery present in the circuit Faraday’s law of induction describes the induced emf

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**EMF Produced by a Changing Magnetic Field, 1**

A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the ammeter deflects

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**EMF Produced by a Changing Magnetic Field, 2**

When the magnet is held stationary, there is no deflection of the ammeter Therefore, there is no induced current Even though the magnet is in the loop

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**EMF Produced by a Changing Magnetic Field, 3**

The magnet is moved away from the loop The ammeter deflects in the opposite direction

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**Faraday’s Law – Statements**

Faraday’s law of induction states that “the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit” Mathematically,

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**Faraday’s Law – Statements, cont**

Remember FB is the magnetic flux through the circuit and is found by 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

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**Faraday’s Law – Example**

Assume a loop enclosing an area A lies in a uniform magnetic field B The magnetic flux through the loop is FB = BA cos q The induced emf is e = - d/dt (BA cos q)

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**Ways of Inducing an emf The magnitude of B can change with time**

The area enclosed by the loop can change with time The angle q between B and the normal to the loop can change with time Any combination of the above can occur

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Lenz’s Law Faraday’s law indicates that the induced emf and the change in flux have opposite algebraic signs This has a physical interpretation that has come to be known as Lenz’s law Developed by German physicist Heinrich Lenz

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Lenz’s Law, cont. Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop The induced current tends to keep the original magnetic flux through the circuit from changing

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Problem 1 A loop of wire in the shape of a rectangle of width w and length L and a long, straight wire carrying a current I lie on a tabletop. Determine the magnetic flux through the loop due to the current I. Suppose the current is changing with time according to I = a + bt, where a and b are constants. Determine the emf that is induced. What is the direction of the induced current in the rectangle?

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Problem 2 The square loop in the figure is made of wires with a total resistance of 20.0 W . It is placed in a uniform T magnetic field directed perpendicular into the plane of the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points A and B is 3.00 m. What is the magnitude and direction of the induced current. [0.0301] A, clockwise

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Problem 3 A 39 turn circular coil of radius 4.80 cm and resistance 1.00 W is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = t t2, where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 5.60 s. [129 mV]

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**Applications of Faraday’s Law – Electric Generator**

An electric conductor, like a copper wire, is moved through a magnetic field, which causes an electric current to flow (be induced) in the conductor

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Generators Electric generators take in energy by work and transfer it out by electrical transmission The AC generator consists of a loop of wire rotated by some external means in a magnetic field

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**Applications of Faraday’s Law – Pickup Coil**

The coil is placed near the vibrating string and causes a portion of the string to become magnetized When the string vibrates at the same frequency, the magnetized segment produces a changing flux through the coil The induced emf is fed to an amplifier

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**Applications of Faraday’s Law – Transformers**

An alternating current in one winding creates a time-varying magnetic flux in the core, which induces a voltage in the other windings Is this a step-up or a step- down transformer?

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Yamanashi maglev The magnetized coil running along the track, called a guideway, repels the large magnets on the train's undercarriage, allowing the train to levitate between 0.39 and 3.93 inches (1 to 10 cm) above the guideway.

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**Sliding Conducting Bar**

A bar moving through a uniform field and the equivalent circuit diagram Assume the bar has zero resistance The work done by the applied force appears as internal energy in the resistor R

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**Sliding Conducting Bar, cont.**

The induced emf is Since the resistance in the circuit is R, the current is

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**Sliding Conducting Bar, Energy Considerations**

The applied force does work on the conducting bar This moves the charges through a magnetic field 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

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**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

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**Induced emf and Electric Fields, cont.**

The emf for any closed path can be expressed as the line integral of E.ds over the path Faraday’s law can be written in a general form:

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**Maxwell’s Equations, Statement**

Faraday’s Law Ampere-Maxwell Law Gauss’s Law in Magnetism Gauss’s Law in Electricity

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**Maxwell’s Equations, Details**

Gauss’s law (electrical): The total electric flux through any closed surface equals the net charge inside that surface divided by eo This relates an electric field to the charge distribution that creates it

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**Maxwell’s Equations, Details 2**

Gauss’s law (magnetism): The total magnetic flux through any closed surface is zero This says the number of field lines that enter a closed volume must equal the number that leave that volume This implies the magnetic field lines cannot begin or end at any point Isolated magnetic monopoles have not been observed in nature

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**Maxwell’s Equations, Details 3**

Faraday’s law of Induction: This describes the creation of an electric field by a changing magnetic flux The law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path One consequence is the current induced in a conducting loop placed in a time-varying B

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**Maxwell’s Equations, Details 4**

The Ampere-Maxwell law is a generalization of Ampere’s law It describes the creation of a magnetic field by an electric field and electric currents The line integral of the magnetic field around any closed path is the given sum

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The Lorentz Force Law Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated F = qE + qv x B This relationship is called the Lorentz force law Maxwell’s equations, together with this force law, completely describe all classical electromagnetic interactions

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**Maxwell’s Equations, Symmetry**

The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes Maxwell’s equations are of fundamental importance to all of science

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