Lecture 14 Magnetic Domains Induced EMF Faraday’s Law Induction Motional EMF

Magnetic Field of a Current Loop – Equation The magnitude of the magnetic field at the center of a circular loop with a radius R and carrying current I is With N loops in the coil, this becomes

Magnetic Field of a Solenoid If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid It is also known as an electromagnet since it acts like a magnet only when it carries a current

Magnetic Field of a Solenoid, 2 The field lines inside the solenoid are nearly parallel, uniformly spaced, and close together This indicates that the field inside the solenoid is nearly uniform and strong The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

Magnetic Field in a Solenoid, 3 The field lines of the solenoid resemble those of a bar magnet

Magnetic Field in a Solenoid, Magnitude The magnitude of the field inside a solenoid is constant at all points far from its ends B = µ o n I n is the number of turns per unit length n = N / ℓ The same result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field in a Solenoid from Ampère’s Law A cross-sectional view of a tightly wound solenoid If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero Apply Ampère’s Law to the blue dashed rectangle

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Magnetic Effects of Electrons – Orbits An individual atom should act like a magnet because of the motion of the electrons about the nucleus Each electron circles the atom once in about every seconds This would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom

Magnetic Effects of Electrons – Orbits, cont The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

Magnetic Effects of Electrons – Spins Electrons also have spin The classical model is to consider the electrons to spin like tops It is actually a quantum effect

Magnetic Effects of Electrons – Spins, cont The field due to the spinning is generally stronger than the field due to the orbital motion Electrons usually pair up with their spins opposite each other, so their fields cancel each other That is why most materials are not naturally magnetic

Magnetic Effects of Electrons – Domains In some materials, the spins do not naturally cancel Such materials are called ferromagnetic Large groups of atoms in which the spins are aligned are called domains When an external field is applied, the domains that are aligned with the field tend to grow at the expense of the others This causes the material to become magnetized

Domains, cont Random alignment, a, shows an unmagnetized material When an external field is applied, the domains aligned with B grow, b

Domains and Permanent Magnets In hard magnetic materials, the domains remain aligned after the external field is removed The result is a permanent magnet In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

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Michael Faraday 1791 – 1867 Great experimental scientist Invented electric motor, generator and transformers Discovered electromagnetic induction Discovered laws of electrolysis

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Faraday’s Experiment – Set Up 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 reads a current and then returns to zero When the switch is opened, the ammeter reads a current 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 The loop has an area A The flux is defined as Φ B = 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 Φ B = Φ B, max = BA When the field is parallel to the plane of the loop, as in b, θ = 90° and Φ B = 0 The flux can be negative, for example if θ = 180° SI units of flux are T. m² = Wb (Weber) DemoDemo

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 DemoDemo

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 B 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 B, can be produced by a change in B, A or θ Since Φ B = 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 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