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Motional Electromotive Force Electromotive force, or most commonly emf is that which tends to cause current (actual electrons and ions) to flow. More formally, emf is the external work expended per unit of charge to produce an electric potential difference across two open-circuited terminals. The emf of a battery can be expressed in terms of electric field established in a conductor: Consider a conductor of length L moving with velocity v perpendicular to a uniform magnetic field B: The magnetic Lorentz force acts on charge (electron) in the conductor: The free charges move towards the end of an conductor and establish an electric field E given by The potential difference between the ends of the conductor is ... and this is the induced emf between the ends of the conductor
For the mutually perpendicular configuration, such as shown in Figure, we obtain: This emf is often called a motional emf, because it depends on the velocity of the conductor moving in the magnetic field. If the conductor forms a part of a circuit: a current will flow. We can discuss this process in terms of the magnetic flux:
The change in magnetic flux associated with conductor’s motion is: The direction of the emf is provided by Lenz’s law: "An induced current is always in such a direction as to oppose the motion or change causing it" (1834) Lenz’s law tells us that the induced emf and the change in flux have opposite signs. the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant. Heinrich Friedrich Emil Lenz (1804-1865)
Lenz's Law extends the principles of energy conservation to situations that involve non-conservative forces in electromagnetism. To see an example, move a magnet towards the face of a closed loop of wire (e.g. a coil or solenoid). An electric current is induced in the wire, because the electrons within it are subjected to an increasing magnetic field as the magnet approaches. This produces an emf that acts upon them. The direction of the induced current depends on whether the north or south pole of the magnet is approaching: an approaching north pole will produce a counter-clockwise current (from the perspective of the magnet), and south pole approaching the coil will produce a clockwise current (see the pervious page). To understand the implications for conservation of energy, suppose that the induced currents' directions were opposite to those just described. Then the north pole of an approaching magnet would induce a south pole in the near face of the loop. The attractive force between these poles would accelerate the magnet's approach. This would make the magnetic field increase more quickly, which in turn would increase the loop's current, strengthening the magnetic field, increasing the attraction and acceleration, and so on. Both the kinetic energy of the magnet and the rate of energy dissipation in the loop (due to Joule heating) would increase. A small energy input would produce a large energy output, violating the law of conservation of energy. Faraday's law of induction The induced electromotive force or emf in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit (1831) Michael Faraday (1791-1867)
moving circuit velocity of the conducting electron the surface element PQQ’P’
Example: a circular loop of radius R rotated at angular speed about one of its diameter in a uniform magnetic field perpendicular to the axis. Calculate emf
Differential Form of Faraday’s Law of Induction Integral form any closed geometric path in space Since C is held fixed, Differential form Note that E cannot in general be expressed as the gradient of a scalar potential!
