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Magnetic field

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**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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**Magnetic field due to currents or magnet**

B due to magnetic moment of electron

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**Forces on charges due to magnetic field (Lorentz force)**

B B Electron beams are deflected by Lorentz force B Horizontal and vertical deflection yoke control the path of electron beams.

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**Force and Torque on a closed circuit**

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

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**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. Magnetic flux density Relative permeability

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Hard disk application 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. (a) Hard disk tracks. (b) Sketch of qualitative shapes of hysteresis curves required for the head and track magnetic materials.

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**Electromagnetic forces on a charge**

1) Electric force (Coulomb force) E F 2) Magnetic force (Lorentz force) B v F

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**Prediction of magnetic field : Biot-Savart law**

Current segment Direction of H-field The magnetic field can be predicted by Biot-Savart’s law with known current distribution.

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Ampere’s law Ampere law facilitates calculation of mangetic field like the Gauss law for electric field.. Unlike Gauss’ law, Ampere’s law is related to line integrals. Ampere’s 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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**Differential form of Ampere’s law**

From the integral form, we will derive the differential form of Ampere’s 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.

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**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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**Example : Surface current**

The direction of magnetic field con be conjectured from the right hand rule.

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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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Example : Torus

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**Faraday’s law 1) Faraday experiment 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. A time-varying flux linking a stationary circuit. A constant magnetic flux with a moving circuit Combination of the above two cases Situations when EMF is generated

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**(1) A time-varying flux linking a stationary circuit.**

+ V -

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**(2) A constant magnetic flux with a moving circuit**

(1) A phenomena observed by a stationary person 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 Direction of induced current Effectively, the motion of bar generates electric field which has the strength of (υ x B) emf = Ed = υBd

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**Example : Hard disk head**

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**(3) Combination of the two**

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**Example : AC generator A simple AC generator**

Observer’s coordinate frame is rotating with the loop. A simple AC generator

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**Example : Eddy current Falling magnet inside a copper tube**

Insulator tube Conductor tube Conductor tube Relative velocity of the copper tube to the magnet.

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Inductance

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**Two important laws on magnetic field**

Current B-field Current generates magnetic field (Biot-Savart Law) Current Time-varying magnetic field generates induced electric field that opposes the variation. (Faraday’s law) Top view B-field Electric field

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

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Concept of inductance 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. Current Ф is the magnetic flux due to the coil wound N times. Ф0 is magnetic flux due to the single turn coil. Magnetic flux : Self inductance is proportional to the square of winding N.

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**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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**Work to move a current loop in a magnetic field**

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. B A I The voltage induced in the current loop hinders the current flow, which should be canceled by an external source.

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**Magnetic energy : Mutual interaction**

The energy is equal to assemble circuits with current Ii. Ij Ii B Magnetic material Energy needed to disintegrate I1, I2,~,In. Energy needed to assemble I1, I2~IN in a free space. (Including self energy)

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Magnetic energy Self energy : The energy needed for the circuit to have a current I flow in spite of the repelling electromotive force from Faraday’s law. (To support current i(t), the current source should provide additional voltage which cancels induced voltage by Faraday’s law.) (Initially, this circuit has a zero current flowing. Then , the current increases to I.)

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**Magnetic energy : two coil system**

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