23.5 self-induction

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23.5 Self-Induction. When the switch is closed, the current does not immediately reach its maximum value Faraday’s Law can be used to describe the effect. Self-Induction,. As the current increases with time, the magnetic flux through the circuit loop also increases with time - PowerPoint PPT Presentation

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  • *23.5 Self-InductionWhen the switch is closed, the current does not immediately reach its maximum valueFaradays Law can be used to describe the effect

  • *Self-Induction, As the current increases with time, the magnetic flux through the circuit loop also increases with timeThis increasing flux creates an induced emf in the circuitThe direction of the induced emf is opposite to that of the emf of the battery The induced emf causes a current which would establish a magnetic field opposing the change in the original magnetic field

  • *Equation for Self-Induction This effect is called self-inductance and the self-induced emf eLis always proportional to the time rate of change of the current

    L is a constant of proportionality called the inductance of the coil It depends on the geometry of the coil and other physical characteristics

  • *Inductance UnitsThe SI unit of inductance is a Henry (H)

    Named for Joseph Henry1797 1878Improved the design of the electromagnetConstructed one of the first motorsDiscovered the phenomena of self-inductance

  • *Inductance of a Solenoid having N turns and Length lThe interior magnetic field is

    The magnetic flux through each turn is

    The inductance is

    This shows that L depends on the geometry of the object

  • *23.6 RL Circuit, IntroductionA circuit element that has a large self-inductance is called an inductorThe circuit symbol is We assume the self-inductance of the rest of the circuit is negligible compared to the inductorHowever, even without a coil, a circuit will have some self-inductance

  • *RL Circuit, AnalysisAn RL circuit contains an inductor and a resistorWhen the switch is closed (at time t=0), the current begins to increaseAt the same time, a back emf is induced in the inductor that opposes the original increasing current

  • *The current in RL CircuitApplying Kirchhoffs Loop Rule to the previous circuit gives

    The current

    where t = L / R is the time required for the current to reach 63.2% of its maximum value

  • *RL Circuit, Current-Time GraphThe equilibrium value of the current is e/R and is reached as t approaches infinityThe current initially increases very rapidlyThe current then gradually approaches the equilibrium value

  • *RL Circuit, Analysis, FinalThe inductor affects the current exponentiallyThe current does not instantly increase to its final equilibrium valueIf there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

  • *Open the RL Circuit, Current-Time GraphThe time rate of change of the current is a maximum at t = 0It falls off exponentially as t approaches infinityIn general,

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  • *23.7 Energy stored in a Magnetic FieldIn a circuit with an inductor, the battery must supply more energy than in a circuit without an inductorPart of the energy supplied by the battery appears as internal energy in the resistorThe remaining energy is stored in the magnetic field of the inductor

  • *Energy in a Magnetic FieldLooking at this energy (in terms of rate)

    Ie is the rate at which energy is being supplied by the batteryI2R is the rate at which the energy is being delivered to the resistorTherefore, LI dI/dt must be the rate at which the energy is being delivered to the inductor

  • *Energy in a Magnetic FieldLet U denote the energy stored in the inductor at any timeThe rate at which the energy is stored is

    To find the total energy, integrate and UB = L I2

  • *Energy Density in a Magnetic FieldGiven U = L I2,

    Since Al is the volume of the solenoid, the magnetic energy density, uB is

    This applies to any region in which a magnetic field exists not just in the solenoid

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  • *Inductance Example Coaxial CableCalculate L and energy for the cableThe total flux is

    Therefore, L is

    The total energy is

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  • *23.8 Magnetic Levitation Repulsive ModelA second major model for magnetic levitation is the EDS (electrodynamic system) modelThe system uses superconducting magnetsThis results in improved energy effieciency

  • *Magnetic Levitation Repulsive Model, 2The vehicle carries a magnetAs the magnet passes over a metal plate that runs along the center of the track, currents are induced in the plateThe result is a repulsive forceThis force tends to lift the vehicleThere is a large amount of metal requiredMakes it very expensive

  • *Japans Maglev VehicleThe current is induced by magnets passing by coils located on the side of the railway chamber

  • *EDS AdvantagesIncludes a natural stabilizing featureIf the vehicle drops, the repulsion becomes stronger, pushing the vehicle back upIf the vehicle rises, the force decreases and it drops back downLarger separation than EMSAbout 10 cm compared to 10 mm

  • *EDS DisadvantagesLevitation only exists while the train is in motionDepends on a change in the magnetic fluxMust include landing wheels for stopping and startingThe induced currents produce a drag force as well as a lift forceHigh speeds minimize the dragSignificant drag at low speeds must be overcome every time the vehicle starts up

  • *Exercises of Chapter 235, 9, 12, 21, 25, 32, 35, 39, 42, 47, 52, 59, 65, 67

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