The magnetic field. The force can be detected by

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Valentine Nicholson
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1 The magnetic field When a field is generated in a volume of space it means that there is a change in energy of that volume, and furthermore that there is an energy gradient so that a force is produced. The force can be detected by 1. The acceleration of an electric charge moving in the field 2.The force on a current-carrying conductor 3.Then torque on a magnetic dipole 4.A reorientation of spins on electrons within certain types of atoms. What cause magnetic field 1.Electrical charge in motion an electrical current flowing in a conductor 2.Permanent magnet- there are the orbital motions and spins of electrons The magnetic field exerts a force on both (1) Current-carrying conductors (2) Permanent magnets 1

Definition of magnetic field strength H the magnetic field H connection The generating electrical current the unit of magnetic field strength: ampere meter (In

A current of 1 ampere passing through a straight 1 meter length of conductor generates a tangential field strength of ¼ π ampere/meter at a radial distance of l

2 Definition of magnetic field strength H the magnetic field H connection The generating electrical current the unit of magnetic field strength: ampere meter (In terms of the generating current) An infinitely long solenoid containing n turns per meter of coil and carrying a current of 1/n amperes. A current of 1 ampere passing through a straight 1 meter length of conductor generates a tangential field strength of ¼ π ampere/meter at a radial distance of l meter. The Biot-savart law (It is a statement experimental observation rather than a theoretical prediction) Enables us to calculate the magnetic field H generated by an electrical current. 2

3 Solenoid A current-carrying conductor 3

4 The law gives the field contribution generated by a current flowing in an elementary length of conductor δ H= 1 iδ l u 4πr 2 i: the current flowing in a conductor δ l :an elemental length of a conductor r: the radius distance u : a unit vector along the radial direction δ H : the contribution to magnetic field at r due to iδl 4

5 -Field due to a long conductor: Determine the H at some point P distant a meters from an infinitely long conductor carrying a i amps. 1 δh iδ u 2 4πr 1 δh iδsin(90 2 4πr δcosα r δα α) rcosα a r δα aδα δ 2 a cosα cos α r cosα icosα δα δh 4πa A B A Bcosθ A B â ABsinθ n 5

6 ( For steady current. Biot-savart law is equivalent to Ampere s Circuital law) ex. Calculate The field at a distance of 10 cm from the conductor when it carries a current of 0.1 A H if 2 i cos d 2 4a i A 2a m a 0.1m and i 0.1A, H 1 2 A or H m A m 6

- Field patterns around current-carrying conductors -The field circulates around a single current-carrying conductor in a direction given by the

7 - Field patterns around current-carrying conductors -The field circulates around a single current-carrying conductor in a direction given by the right-hand corkscrew rule. - If we look along the conductor in the direction of the conventional current, the magnetic field circulates in a clockwise direction. 7

8 8

- The line of force leave the N pole and return at the S outside the magnet.

9 - In a bar magnet, the field emerges from one end of the magnet - north pole (N) of a magnet as a source of magnetic field H. While a south pole (S) behaves as a field sink. - The line of force leave the N pole and return at the S outside the magnet. (Whether such poles have any real existence is debatable) - The strength of the magnetic field is proportional to the density of the line of force. 9

10 - Notice that H produced by a bar magnet that of a solenoid. In particular, the magnetic field lines within the bar magnet run in the opposite direction to the field lines within the solenoid. - It can be explained because the bar magnet has a magnetization M, while the solenoid does not, and this M leads to the generation of a magnetic dipole which acts as a source and sink for magnetic field. Ampere s circuital law ( How can we calculate the strength of a magnetic field generated by an electrical current?) - The magnetic field generated by an electrical circuit. (According to Ampere) Depended on the shape of the circuit ( conduction path ) the current carried - By assuming that each circuit is made up of an infinite number of current elements each contributing to the field, and by summing or integrating these contribution at a point to determine the field. 10

11 line vector nteracting along a closed path around the conductor at a distance r H d 2πrH i H i 2πr According above equations, Ampere s law = Biot-savart law. 11

12 Magnetic induction (B) (How does a medium respond to magnetic field?) - Magnetic induction B, sometime call the flux density. - When a magnetic field H has been generated in a medium (in accordance with Ampere s law), the response of the medium is its magnetic induction B. (All media will respond with some induction.) - Permeability of medium: the relation between magnetic induction (B) and magnetic field (H) B unit: Webers meter B f (F) magnetic induction 2 Tesla (The Weber is the amount of magnetic flux) the force on a moving electric charge or electric current - A magnetic induction B of 1 Tesla generates a force of 1 Newton per meter on conductor carrying a current of 1 Ampere perpendicular to the direction of the induction 12

13 magnetic field H magnetic induction (B) magnetization M of There are two contributions to magnetic the medium H induction M In free space B H 0 In many medium, B is a linear function of In particular in free space H μ 0 : the permeability of free space (universal constant) A H : m amp meter B : tesla V s volt second 2 2 m meter 0 B H V s 2 m A m If the value of B in free space is known, then H in free space is immediately known from this relationship V s volt second m A meter amp H m 7 heneries meter H m 13

14 ferromagne ts ferrimagne ts nor is it even a single - valued function of H - However in other media, is no longer a linear function of H. 1.In paramagnets and diamagnets μ is constant over a considerable range B μh of values of H 2.In ferromagnet μ varies rapidly with H is not necessarily a constant B - A field gives rise to magnetic induction B H A m (tesla) in a medium with permeability. μ H m 14

15 Magnetic flux (Φ) -Whenever a magnetic field is present in free space, there will be a magnetic flux (Φ). - Unit: weber -The weber is the amount of magnetic flux which when reduced uniformly to zero in one second produces an e.m.f. of one volt in a one-turn coil of conductor through which the flux passes. -The amount of flux generated by a given field strength depends on the properties of the medium and varies from one medium to another. 15

Force per unit length on a current-carrying conductor in a magnetic field (The unit of magnetic induction has been defined in terms of the force exerted on a current-carrying conductor.

16 Force per unit length on a current-carrying conductor in a magnetic field (The unit of magnetic induction has been defined in terms of the force exerted on a current-carrying conductor. This will now be generalized to obtain the force F on a current-carrying conductor in a magnetic induction B) B f (F) The force exerted on current-carrying conductor The force per meter on a conductor carrying a current i in the direction of the unit vector l caused by a magnetic induction B. F il B i : conductor carrying a current l :current direction - In free space, F il H o F o i i 2a 1 2 i 1 a i 2 同相 : 相互吸引異相 : 相互排斥 If two long wires are arranged parallel at a distance of a meter apart and carry currents of i 1 and i 2 amps the force per meter exerted by one wire on the other is: o F i1i 2a 2

Electromagnetic induction - (can the magnetic field generate an electrical current or voltage in return?) - When the magnetic flux linking an electric circuit changes an e.m.f. is induced and this phenomenon is called electromagnetic induction.

17 Electromagnetic induction - (can the magnetic field generate an electrical current or voltage in return?) - When the magnetic flux linking an electric circuit changes an e.m.f. is induced and this phenomenon is called electromagnetic induction. - Faraday s law: the voltage induced in an electrical circuit is proportional to the rate of change of magnetic flux linking the circuit. d d V ( ) N : magnetic flux is equal to the induced e.m.f. dt dt Where is the magnetic flux passing through a coil of N turns and d is dt the rate of change of flux 17

18 Lenz s law - The induced voltage is in a direction which oppose the flux change producing it. ( 電動勢所產生的電流往反抗磁通量變化的方向流動 ) - Magnetic flux: ϕ (weber) (since the magnetic induction is the flux density) B A db V NA dt - Important result (an electrical current can be generated by a time-depend) - Ex. What is the voltage induced in a 50 turn coil area 1 cm 2 when the magnetic induction linking it changes uniformly from 3 test to zero in 0.01 seconds? db (50)(110 V NA dt )(3) 1.5 volts

The magnetic dipole (What is the most elementary unit of magnetism) A circular loop of a conductor carrying an electric current, which can generate a magnetic field.

19 The magnetic dipole (What is the most elementary unit of magnetism) A circular loop of a conductor carrying an electric current, which can generate a magnetic field. A circular current loop can be considered the most elementary unit of magnetism. If a current loop has area A and carries a current i, then its magnetic dipole moment is m=ia. The units of magnetic moment: A m 2 (amp meter 2 ) B tries align the dipole, so that the moment m lies parallel to the induction The torque on a magnetic dipole of moment m in a magnetic induction B is then simply In free space m B m H o (-This mean that B tries to align the dipole so that the moment m lies parallel to the induction) 19

20 - The energy of the dipole moment m in the presence of a magnetic induction (If no frictional forces are operating, the work done by the turning force will be conserved) E m B In free space E m H o The field produced by a current loop is identical in form to the field produced by calculation from two hypothetical magnetic poles of strength separated by a distance l. i M A S -P N l +P P: magnetic pole strength l: distance

III.Sources of Magnetic Fields - Ampere s Law - solenoids

Magnetism I. Magnetic Field - units, poles - effect on charge II. Magnetic Force on Current - parallel currents, motors III.Sources of Magnetic Fields - Ampere s Law - solenoids IV.Magnetic Induction -