Magnetostatics III
Magnetization All magnetic phenomena are due to motion of the electric charges present in that material. A piece of magnetic material on an atomic scale have tiny currents due to electrons orbiting around nuclei and electrons spinning about their axes. But for macroscopic purposes, these current loops are so small that we may treat them as magnetic dipoles. Actually, they cancel each other out because of the random orientation i of the atoms.
But when a magnetic field is applied, a net alignment of these magnetic dipoles occurs, and the medium becomes magnetically polarized, or magnetized. Electric polarization, generally occurs in the direction of E. However magnetization can be parallel to B (paramagnets) and also can be opposite to B (diamagnets). A few substances (called ferromagnets, example: iron) retain their magnetization even after the external field has been removed.
For such materials the magnetization is not determined by the present field but by the whole magnetic "history" of the object. In the presence of a magnetic field, matter becomes magnetized because tiny dipoles align them self along some direction. M = magnetic dipole moment per unit volume. M is called the magnetization
Ampere s Law: Electric currents create magnetic fields.
Lorentz Force: Charges moving in a magnetic field experience an electromagnetic force.
Faraday s Law of Induction: A changing magnetic field creates an electric field.
Lenz s Law: Induced electric currents act so as to oppose the motion that caused them.
The Biot Savart s Law Biot Savart s law states that: dh 1 d I Rˆ 4 2 R A/m where: dh = differential magnetic field di = differential current element
Magnetic Fields Charges induce electric field q 3 q 2 q 1 Q F F E Q Source charges Test charge
( v B)
MagneticForce between Two Parallel Conductors Force per unit length on parallel current carrying conductors is: F' ˆ 1 y I 0 1 I 2d where F 1 = F 2 (attract each other with equal force) 2
Relative permeability Recall how field in vacuum capacitor is reduced when dielectric medium is inserted; alwaysreduction reduction, whether medium is polar or non polar: Evac E B rbvac r is the analogous expression when magnetic medium is inserted din the vacuum solenoid. Complication: the B field can be reduced orincreased, depending on the type of magnetic medium
Magnetic Forces and Torques The electric force F e per unit charge acting on a test charge placed at a point in space with electric field E. When a charged particle moving with a velocity v passing through that point in space, the magnetic force F m is exerted on that charged particle. F m Qv B N where B = magneticflux density (Cm/s or Tesla T)
Magnetic Forces and Torques If a charged particle is in the presence of both an electric field E and magnetic field B, the total electromagnetic force acting on it is: F F e F m QE Qv B Q E v B (Lorentz force)
Combined E and B fields F qe q ( v B) Acts on any particle, whether moving or in rest Acts on moving particle only
Lorentz s law: Magnetic force on a moving charge: F mag Q( v B) Right hand rule Total force on a charge: F Q( ( E v B) )
Magnetic Torque on a Current Carrying Loop Applied force vector F and distance vector d are used to generate a torque T T = d F (N m) Rotation direction is governed by right hand hand rule.
Electric Generators In a motor we have seen that a current loop in an external magnetic field produces a torque on the loop. In a generator we ll see that a torque on a current loop inside a magnetic field produces a current. In summary: Motor: Current + Magnetic field Torque Generator: Torque + Magnetic field Current Turbines in a power plant are usually rotated either by a waterfall or by steam created heat produced dfrom nuclear power or the burning of coal. As the turbines rotate, current loops turn through a magnetic field to generate electricity. This process converts mechanical energy into electrical energy. The simplest form of an electric generator is called an alternating current (AC) generator. The current produced by an AC generator switches directions every time the wire inside of it is rotated through a half turn. In the United States, generator generally have a frequency of 60 Hz, which means the current switches direction 120 times every second! A graph of the current output from an AC generator produces a sinusoidal id curve due to the periodic nature of the generator s rotation. ti
Electric Generator B I induced d As a turbine turns (due to some power source like coal) a current loop (purple) is rotated inside a magnetic field. The field is static but as the loop turns as the number of field lines poking through it changes. Thus we have a changing flux and a corresponding induced emf and current. The pic shows a loop just after it was horizontal (perpendicular to the field). The flux is decreasing since the loop is becoming more vertical. Since fewer field lines are entering the loop, the induced current is in a direction to produce more field lines downward. Just prior to this, as the loop was approaching horizontal, the number of field lines inside it was increasing, so the current was in the other direction to oppose this change. The current changes direction twice with each turn whenever the loop is horizontal. The result here is AC, but (direct current) DC motors exist as well in which current only flows in one direction.
The Biot-Savart Law The magnetic field of a steady current 0 I Rˆ I d B r 0 ( ) d 2 2 4 R 4 R Rˆ Biot Savart law R rp P 0 10 7 2 4 N / A permeability of free space d 1T( teslas ) 1 N /( Am ) units 21
Biot Savart Law The analogue of Coulomb s Law is the Biot Savart Law r r r db(r) Consider a current loop (I) For element tdl there is an d' x( r r' associated element field db db( r) 3 4 r r' db perpendicular to both dl and r r same 1/(4r 2 ) dependence o is permeability of free space defined as 4 x 10 7 Wb A 1 m 1 o d' x( r r' ) B ( r ) 3 4 Integrate to get B S Law r r' O r dl o )
The Biot Savart s Law To determine the total H: H 1 dl Rˆ 2 4 R l A/m
The Biot Savart Savart s Law Biot Savart s law may be expressed in terms of distributed current sources. H H 1 4 1 4 S v J ˆ s R ds 2 R J Rˆ R 2 dv for a surface current for a volume current
B S Law examples (1) Infinitely long straight conductor I dl dl and r, r in the page db is out of the page B forms circles centred on the conductor Apply B S Law to get: r z r r O db r r sin = cos 2 2 1/2 r z 1/2 B o I 2 r B
Ampere s law (Electric currents create magnetic fields) B ( r ) J ( r ) Ampere s law in differential form o ( B) da B d o J ( r ) da I o enc B d o I enc Ampere s law in integral form Electrostatics : Magnetostatics : coulomb law Biot Savart law Gauss' s law Ampere' s law 26
Application of Ampere s law A toroidal coil consists of a circular ring, or donut, around which a long wire is wrapped. The winding is uniform and tight enough so that each turn can be considered aclosed loop. The cross sectional sectional shape of the coil is immaterial. Imade it rectangular in figure a for the sake of simplicity, it but it could just as well becircular or even some weird id asymmetrical form, as in figure b, just as long as the shape remains the same all the way around the ring. In that case it follows that the magnetic field of the toroid is circumferential at all points, both inside and outside the coil. a b 27
Application of Ampere s law Ampere s loop from Ampere' B d I o enc s law 2 B ˆ sdˆ oni 0 2sB oni NI o oni B B ˆ 2s 2s NI o ˆ, for points inside the coil B 2s 0, for points outside the coil 28
Application of Ampere s law 29
30
31
32
Comparison of Magnetostatics and Electrostatics 1 E, Gauss' s law B 0, no name 0 B 0J, Ampere' s E 0, no name F Q(E v B), Force law law 33