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1 EMLAB Magnetic field

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2 EMLAB Generation of magnetic field A charged particle in motion generates magnetic field nearby. In the same way, currents generate magnetic field nearby.

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3 EMLAB Magnetic field due to currents or magnet B due to magnetic moment of electron

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4 EMLAB Forces on charges due to magnetic field (Lorentz force)B B B Electron beams are deflected by Lorentz force Horizontal and vertical deflection yoke control the path of electron beams.

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5 EMLAB Force and Torque on a closed circuit Current loops in a magnetic field experience torque, and are rotated until the plane of loops are perpendicular to the applied B. F F B Angular acceleration is proportional to the applied torque. Torque is proportional to the product of radius and force. If the sum of torques due to A and B has nonzero value, the seesaw is rotated.

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6 EMLAB Relative permeability Magnetic flux density Magnetic field and magnetic flux density Arrows represent magnetic field due to orbiting electrons. The orbits of electrons are aligned due to external magnetic field.

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7 EMLAB (a) Hard disk tracks. (b) Sketch of qualitative shapes of hysteresis curves required for the head and track magnetic materials. The magnetic head aerodynamically flies over the disk surface at a distance above it of only about 1mm while following the surface profile. In the figure, the surface profile is shown as ideally flat, which in practice is not the case. Hard disk application

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8 EMLAB Electromagnetic forces on a charge 1) Electric force 2) Magnetic force F E F B v (Lorentz force) (Coulomb force)

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9 EMLAB Prediction of magnetic field : Biot-Savart law Direction of H-field Current segment The magnetic field can be predicted by Biot-Savarts law with known current distribution.

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10 EMLAB Amperes law Ampere law facilitates calculation of mangetic field like the Gauss law for electric field.. Unlike Gauss law, Amperes law is related to line integrals. Amperes law is discovered experimentally and states that a line integral over a closed path is equal to a current flowing through the closed loop. In the left figure, line integrals of H along path a and b is equal to I because the paths enclose current I completely. But the integral along path c is not equal to I because it does not encloses completely the current I.

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11 EMLAB From the integral form, we will derive the differential form of Amperes law. Line integrals from these adjacent currents add up to zero. Line integrals over a closed path is equal to the sum of line integrals over infinitesimally small loops. Differential form of Amperes law

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12 EMLAB Example- Coaxial cable The direction of magnetic fields can be found from right hand rule. The currents flowing through the inner conductor and outer sheath should have the same magnitude with different polarity to minimize the magnetic flux leakage

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13 EMLAB Example : Surface current The direction of magnetic field con be conjectured from the right hand rule.

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14 EMLAB Example : Solenoid The direction of magnetic field con be conjectured from the right hand rule. If the length of the solenoid becomes infinite, H field outside becomes 0.

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15 EMLAB Example : Torus

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16 EMLAB N S Electromotive force (emf) (-) sign explains the emf is induced across the terminals of the coil in such a way that hinders the change of the magnetic flux nearby. 1.A time-varying flux linking a stationary circuit. 2.A constant magnetic flux with a moving circuit 3.Combination of the above two cases Situations when EMF is generated Faradays law 1) Faraday experiment

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17 EMLAB +V-+V- (1) A time-varying flux linking a stationary circuit. Time varying

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18 EMLAB (2) A constant magnetic flux with a moving circuit (1) A phenomena observed by a stationary person Direction of induced current Due to the motion of a conducting bar, the charges in it moves in the (+y) direction. The moving charges experience Lorentz force such that 1.Effectively, the motion of bar generates electric field which has the strength of (υ x B) 2.emf = Ed = υBd

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19 EMLAB Example : Hard disk head

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20 EMLAB (3) Combination of the two

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21 EMLAB Example : AC generator A simple AC generator Observers coordinate frame is rotating with the loop.

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22 EMLAB Example : Eddy current Relative velocity of the copper tube to the magnet. Falling magnet inside a copper tube Insulator tubeConductor tube

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23 EMLAB Inductance

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24 EMLAB Two important laws on magnetic field Current generates magnetic field (Biot-Savart Law) Time-varying magnetic field generates induced electric field that opposes the variation. (Faradays law) Current B-field Top view Electric field B-field

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25 EMLAB Current Magnetic flux : Magnetic flux B in a solenoid with N turn coil

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26 EMLAB Concept of inductance Current Magnetic flux : Ф is the magnetic flux due to the coil wound N times. Ф 0 is magnetic flux due to the single turn coil. Self inductance is proportional to the square of winding N. The change of magnetic flux intensity due to changing current generates electromotive force. The proportionality constant between the emf and current is called a inductance.

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27 EMLAB Mutual Inductance (1) When the secondary circuit is open The current flowing through the primary circuit generates magnetic flux, which influences the secondary circuit. Due to the magnetic flux, a repulsive voltage is induced on the secondary circuit.

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28 EMLAB Work to move a current loop in a magnetic field I BABA If we want to move a current loop with I flowing in a region with a magnetic flux density B, energy should be supplied from an external source. The voltage induced in the current loop hinders the current flow, which should be canceled by an external source.

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29 EMLAB B The energy is equal to assemble circuits with current I i. Magnetic energy : Mutual interaction IiIi IjIj Energy needed to assemble I 1, I 2 ~I N in a free space. Energy needed to disintegrate I 1, I 2,~,I n. Magnetic material (Including self energy)

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30 EMLAB Magnetic energy (Initially, this circuit has a zero current flowing. Then, the current increases to I.) (To support current i(t), the current source should provide additional voltage which cancels induced voltage by Faradays law.) Self energy : The energy needed for the circuit to have a current I flow in spite of the repelling electromotive force from Faradays law.

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31 EMLAB Magnetic energy : two coil system

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