Chapter 8. Magnetic forces, materials, and inductance

8.1 Force on a moving charge 1) Electric force (Coulomb force) E F 2) Magnetic force (Lorentz force) B v F

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

Examples CRT use Lorentz force to steer electrons emitted from the cathode.

8.2 Force on a differential current element B B

B

Example 8.1

8.3 Force between differential current elements

8.4 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.

Example of torques due to magnetic forces Rotation axis Angular acceleration is proportional to the applied torque.

Calculation of torques on a closed loop current Any closed current loops can be decomposed into a set of infinitesimally small square current loops. If the torque due to a small square current loops is known, torque on arbitrarily shaped current loop can be evaluated straightforwardly by simple addition. B is assumed to be constant on each segment of a small current loop. m : magnetization of the current loop : position vectors from the origin at the center of the current loop. S : area of the small loop : normal vector of the area Center of the current loop.

Proof :

Simple motor Magnet Neodymium magnet

8.5 The nature of magnetic materials B In a magnetic field, orbits of electrons are influenced by the magnetic torques. F F - Electron : 1) orbital spin 2) electron spin B F F

Diamagnetic : 반자성체  <1 Paramagnetic : 상자성체   1 The mount of the rotation of electron orbits due to torques makes up material properties. Paramagnetic material is influenced by external magnetic field, tend to align its magnetic moment along the external field. Diamagnetic : 반자성체  <1 Paramagnetic : 상자성체   1 Magnetic moments of an electron pair in valence band cancel each other. Magnetic moment of an unpaired electron is not canceled.

B-H curve Diamagnetic : 반자성체  <1 Paramagnetic : 상자성체   1 Ferromagnetic : 강자성체  >> 1 Antiferromagnetic : Ferrimagnetic : Ferrite (small ) B H

Magnetization and demagnetization Magnetic polarization due to magnetization of electron orbits. External magnetic field H not applied H applied

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.

8.7 Magnetic boundary conditions 1) Normal component 2) Tangential component If either of the two medium is conductor, Jh is nonzero with h→0.

8.8 Magnetic circuit

Example 8.9

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. Situations when EMF is generated A time-varying flux linking a stationary circuit. A constant magnetic flux with a moving circuit Combination of the above two cases

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

(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 (v x B) emf = Ed = vBd

Analogy to inertial force An elevator which is accelerated in downward direction. Fi : Inertial force Acceleration of a ball observed by moving Pikachu. Acceleration of a ball observed by a stationary rabbit.

(3) Combination of the two

Example : AC generator A simple AC generator

Example : Hard disk head

Concept of inductance Current Magnetic flux :

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.

9.9 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.)

Work to move current loop 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.

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)

8.9 Magnetic energy The energy needed to assemble current loops with constant current sources Ii . Ij Ii B Wi,j Energy needed to disassemble Energy needed to assemble

Magnetic energy by Field variables If the integration surface extend to infinity, the second term banishes.

Inductance calculation by Energy The induced voltage is generated by the circuit itself. - V + I

Mutual Inductance Field equations are useful in deriving mutual inductances. circuit 1 circuit 2 The current flowing through the circuit 1 generates magnetic flux, which induces emf in the second coil.

Example 8.9