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Law Faraday's of Induction This is generally expressed as: an induced emf is produced in the secondary circuit by the changing magnetic field The actual existence of the magnetic field is not sufficient to produce the induced emf, the field must be changing Magnetic Flux To express Faraday's finding mathematically, the magnetic flux is used The flux depends on the magnetic field and the area: r r B = B dA Flux and Induced emf An emf is induced in a circuit when the magnetic flux through the surface bounded by the circuit changes with time This summarizes Faraday's experimental results Faraday's Law Statements Faraday's Law of Induction states that the emf induced in a circuit is equal to the time rate of change of the magnetic flux through the circuit Mathematically, d B =- dt Faraday's Law Statements, cont If the circuit consists of N identical and concentric loops, and if the field lines pass through all loops, the induced emf becomes d B = -N dt The loops are in series, so the emfs in the individual loops add to give the total emf Faraday's Law Example Assume a loop enclosing an area A lies in a uniform magnetic field The magnetic flux through the loop is B = B A cos The induced emf is d = - ( BA cos ) dt r r B = B dA A circular loop of wire is held in a uniform magnetic field, with the plane of the loop perpendicular to the field lines. Which of the following will not cause a current to be induced in the loop? 1. 2. 3. 4. crushing the loop rotating the loop about an axis perpendicular to the field lines keeping the orientation of the loop fixed and moving it along the field lines pulling the loop out of the field d = - ( BA cos ) dt Ways of Inducing an emf The magnitude of the field can change with time The area enclosed by the loop can change with time The angle between the field and the normal to the loop can change with time Any combination of the above can occur d = - ( BA cos ) dt Figure 23.7 shows a graphical representation of the field magnitude versus time for a magnetic field that passes through a fixed loop and that is oriented perpendicular to the plane of the loop. The magnitude of the magnetic field at any time is uniform over the area of the loop. Rank the magnitudes of the emf generated in the loop at the five instants indicated, from largest to smallest. 1. 2. 3. c, d = e, b, a a, d = e, b, c a, d, c, e, b d = - ( BA cos ) dt Applications of Faraday's Law Pickup Coil The pickup coil of an electric guitar uses Faraday's Law The coil is placed near the vibrating string and permanent magnet causes a portion of the string to become magnetized When the string vibrates at some frequency, the magnetized segment produces a changing flux through the coil The induced emf is fed to an amplifier Motional emf A motional emf is one induced in a conductor moving a through magnetic field The electrons in the conductor experience a force that is directed along l r r r FB = qv B Motional emf, cont Under the influence of the force, the electrons move to the lower end of the conductor and accumulate there As a result of the charge separation, an electric field is produced inside the conductor The charges accumulate at both ends of the conductor until they are in equilibrium with regard to the electric and magnetic forces Motional emf, final For equilibrium, q E = q v B or E=vB A potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field If the direction of the motion is reversed, the polarity of the potential difference is also reversed 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 Sliding Conducting Bar, cont The induced emf is d B dx = -Bl = -Blv =- dt dt Since the resistance in the circuit is R, the current is Blv I= = R R You wish to move a rectangular loop of wire into a region of uniform magnetic field at a given speed so as to induce an emf in the loop. The plane of the loop must remain perpendicular to the magnetic field lines. In which orientation should you hold the loop while you move it into the region of magnetic field so as to generate the largest emf? 1. 2. 3. with the long dimension of the loop parallel to the velocity vector with the short dimension of the loop parallel to the velocity vector either way because the emf is the same regardless of orientation = -Blv 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, if velocity is constant 2 = Fappv = ( I lB ) v = R In Active Figure 23.11, a given applied force of magnitude Fapp results in a constant speed v and a power input P. Imagine that the force is increased so that the constant speed of the bar is doubled to 2v. Under these conditions, what are the new force and the new power input P? 1. 2. 3. 4. 2F and 4F and 2F and 4F and 2P 2P 4P 4P Problem 23.14. A conducting rod of length moves on two horizontal, frictionless rails as shown in Figure P23.12. If a constant force of 1.00 N moves the bar at 2.00 m/s through a magnetic field that is directed into the page, (a) what is the current through the 8.00- resistor R? (b) What is the rate at which energy is delivered to the resistor? (c) What is the mechanical power delivered by the force ?

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