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29.3 Lenz's Law • Is a convenient alternative method for determining the direction of an induced emf or current • is not independent – can be derived from Faraday's law • due to the Russian scientist H.F.E Lenz (1804-1865) Lenz's law states: The direction of any magnetic induction effect is such as to oppose the cause of the effect. The “cause” may be changing flux • through a stationary circuit due to a changing B-field • due to motion of the conductors that make up the circuit M. Moodley, 2009 1 If the flux in a stationary circuit changes • the induced current sets up a B-field of its own • within the area bounded by the circuit, this field is opposite to the original field if the original field is increasing • but is in the same direction as the original field if the latter is decreasing the induced current opposes the change in flux through the circuit (not the flux itself) M. Moodley, 2009 2 If the flux change is due to motion of conductors • the direction of the induced current in the moving conductor is such that the direction of the B-field force on the conductor is opposite in direction to its motion the motion of the conductor, which caused the induced current, is opposed In all cases: the induced current tries to preserve the status quo by opposing motion or a change of flux M. Moodley, 2009 3 In the slidewire generator, the induced current in the loop causes an additional B-field in the area bounded by the loop. The direction of the induced current is counter-clockwise. The magnitude of the uniform B-field is increasing, and the resulting induced current is in the clockwise direction M. Moodley, 2009 4 M. Moodley, 2009 5 In which direction is the current induced in the loop for each situation? M. Moodley, 2009 6 29.4 Motional Electromotive Force To understand the origin of the induced emf we consider the magnetic forces on mobile charges in the conductor: (lets look at a rod moving in a uniform B-field) • a charge +q in the rod experiences a magnetic force • the magnetic force causes free charges in the rod to move and excess positive charges collect at the upper end a and negative charges at the lower end b this creates an electric field E within the rod, directed from a toward b and an electric force in the same direction M. Moodley, 2009 7 • charge continues to accumulate at the ends until E becomes large enough for the downward E-force to cancel the upward B-force (charges are in equilibrium) • the magnitude of the potential difference is then given by (using ) • with point a at a higher potential than point b M. Moodley, 2009 8 Now suppose the moving rod slides along a stationary U- shaped conductor, forming a complete circuit • no magnetic force acts on the charges in the stationary U-shaped conductor • but the charge that was near points a and b redistributes itself along the stationary conductors, creating an E-field within them • this E-field creates a current in the counter-clockwise direction M. Moodley, 2009 9 • the moving rod has become a source of electromotive force • within it, charge moves from lower to higher potential • and in the remainder of the circuit, charge moves from higher to lower potential • this emf is called a motional electromotive force : • corresponds to a force per unit charge of magnitude vB acting for a distance L along the moving rod Can generalise this for a conductor of any shape moving in any B-field: for an element dl of conductor we have For any closed conducting loop, the total motional emf is ( an alternative formulation of Faraday's law ) M. Moodley, 2009 10 Example: Calculating motional emf Suppose the length L in the figure is 0.10 m, the velocity v is 2.5 m/s, the total resistance of the loop is 0.030 Ω , and B is 0.60 T. Find E (the induced emf), the induced current, and the force acting on the rod. M. Moodley, 2009 11 Example: The Faraday disk dynamo A conducting disk with radius R, lies in the xy-plane and rotates with constant angular velocity ! about the z-axis. The disk is in a uniform, constant B-field parallel to the z-axis. Find the induced emf between the centre and the rim of the disk. M. Moodley, 2009 12 29.5 Induced Electric Fields An induced emf also occurs when there is a changing flux through a stationary conductor. Consider the following situation of a long, thin solenoid with cross-sectional area A and n turns per unit length: G measures current in the loop M. Moodley, 2009 13 • the current I in the windings set up a B-field along the solenoid axis with magnitude • taking the area vector A to point in the same direction as B, the magnetic flux through the loop is • if the solenoid current I changes with time, the flux changes and according to Faraday's law the induced emf in the loop is • if the total resistance in the loop is R, the induced current in the loop I' is M. Moodley, 2009 14 But what force makes the charges move around the loop? • it cannot be due to the B-field • it is due to an induced E-field in the conductor caused by the changing magnetic flux • this E-field is not conservative • the line integral, representing the work done by the induced E-field per unit charge is equal to the induced emf: • Faraday's law can therefore be restated as • is only valid if the path around which we integrate is stationary M. Moodley, 2009 15 Example: Induced electric fields Suppose the long solenoid is wound with 500 turns per metre and the current in the windings is increasing at the rate of 2 100 A/s. The cross-sectional area of the solenoid is 4.0 cm . a) Find the magnitude of the induced emf in the wire loop outside the solenoid. b) Find the magnitude of the induced electric field within the loop if its radius is 2.0 cm. M. Moodley, 2009 16

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champ électrique, capacité d'un condensateur, champ magnétique, Jean-Didier Legat, Charles Trullemans

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posted: | 3/7/2011 |

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