Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Electrical resistance and conductance wikipedia , lookup
State of matter wikipedia , lookup
Field (physics) wikipedia , lookup
History of electromagnetic theory wikipedia , lookup
Maxwell's equations wikipedia , lookup
Condensed matter physics wikipedia , lookup
Magnetic field wikipedia , lookup
Neutron magnetic moment wikipedia , lookup
Electromagnetism wikipedia , lookup
Magnetic monopole wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Superconductivity wikipedia , lookup
Ch. 28: Sources of Magnetic Fields Electric Currents Create Magnetic Fields A long, straight wire A current loop A solenoid Slide 24-14 Biot-Savart Law • Current produces a magnetic field • The Biot-Savart law gives the magnetic field arising from an infinitesimal current element: ~ ⇥ r̂ µ0 I dL ~ dB = 4⇡ r2 • Permeability of free space: µ0 = 4⇡ ⇥ 10 7 T m/A • Integrating along the wire: ~ = B Z ~ ⇥ r̂ µ0 I dL 4⇡ r2 ~ = B Z ~ ⇥ r̂ µ0 I dl 4⇡ r2 At location P, what is the direction of the infinitesimal contribution dB created by the current in dl? A) Up the page B) Directly away from dl (in the plane of the page) C) Into the page D) Out of the page E) Some other direction ~r ~ = B Z ~ ⇥ r̂ µ0 I dl 4⇡ r2 P What is the magnitude of ~ ⇥ r̂ dl ? r2 θ ~r dl Origin a) dl sin ✓ r2 c) dl cos ✓ r2 b) dl sin ✓ r dl cos ✓ d) r3 e) dl r2 Magnetic field of long straight wire ~ = B sin(⇥) = sin( Z ⇥) = p µ0 I a sin ✓ Bz = dx 2 4⇡ a r sin ✓ = sin(⇡ ✓) = d/r • as a goes to infinity: µ0 I Bz ! 2⇡d x Z ~ ⇥ r̂ µ0 I dl 4⇡ r2 x2 + y 2 µ0 I ) Bz = 4⇡ Z a a d dx (d2 + x2 )3/2 µ0 I 2a p Bz = 4⇡ d d2 + a2 Clicker Question CT 32.14b What is the direction of the Force acting on the Red Wire? I A) Up B) Right C) Left D) Into the Page E) Out of the Page ©University of Colorado, Boulder (2008) I CT 32.7a A long straight wire carries current I out of the page. An electron travels towards the wire from the right. Which way is the force on the electron? I - e v A: B: C: ← D: ↓ E: 0 ©University of Colorado, Boulder (2008) B field due to a current loop • Only component of dB in x direction doesn’t cancel out µ0 I dL dBx = cos 2 2 4⇥ x + a B= µ0 Ia 2 ( 2 2 x +a 2 32 ) • For large distances (x >> a), this reduces to B= µ0 Ia 2 2x 3 ~ = B Z ~ ⇥ r̂ µ0 I dL 4⇡ r2 CT 32.16 What is the direction of the B- Field at point P? A: B: C: 0 D: → E: Other ©University of Colorado, Boulder (2008) Magnetic dipoles • A small current loop constitutes a magnetic dipole. • Magnitude of B decreases with distance as 1/x3 • Its dipole moment is µ = IA, with A the loop area. • For an N-turn loop, µ = NIA. • The direction of the dipole moment vector is perpendicular to the loop area. • The fields of electric and magnetic dipoles are similar far from their sources, but differ close to the sources. Clicker Question CT 32.14c Two loops of wire have current going around in the same direction. The magnetic forces on the loops are: i2 A: Attractive B: Repulsive C: Net force is zero. i1 ©University of Colorado, Boulder (2008) Ampère’s law • From Zthe Biot-Savart law, ~ ~ = B µ0 I dL ⇥ r̂ 4⇡ r2 • Ampere’s law can be derived I ~ = µ0 Ienc ~ · dl B • For steady currents • The integral is taken around any closed loop, and Ienc is the current encircled by that loop. • Useful only for current distributions with symmetry Field due to a long cylindrical conductor outside conductor: Iencl=I µ0 I 2 Br = µ0 I =) B = 2 r inside conductor: Iencl = J⇡r 2 , J = I ⇡R2 2 µ0 Ir r 2 2 Br = µ0 J r = µ0 I 2 =) B = 2 R2 R Solenoids • A solenoid is a long, tightly wound coil of wire. • When a solenoid’s length is much greater than its diameter, the magnetic field inside is nearly uniform except near the ends, and the field outside is very small. . Clicker Question • In the ideal limit of an infinitely long solenoid, the field inside the solenoid is uniform everywhere, and the field outside is zero. • Application of Ampère’s law shows that the field of an infinite solenoid is B = µ0nI, where n is the number of turns per unit length I ~ = BL = µ0 nIL ~ · dl B B = µ0 nI Magnetism in matter • Magnetism in matter arises from atomic current loops associated with valence electrons having orbital angular momentum and spin. Classical picture of magnetic dipole moment arising from orbiting electron Magnetism in matter • In ferromagnetic materials like iron, strong interactions among individual magnetic dipoles result in large-scale magnetic properties, including strong attraction to magnets. Inducing a Magnetic Moment in a Piece of Iron Electromagnet • Use current to induce magnetic moment in iron Magnetism in matter • Paramagnetic materials exhibit much weaker magnetism. • Diamagnetic materials respond oppositely, and are repelled by magnets. B = Bvac (1 + ) Hysterisis • Consider again a charged particle moving near a long wire with current I. Q v I • Recall the Lorentz force is given by ~ F~ = q~v ⇥ B • What frame of reference of reference do we use to determine the velocity? • Answer: Same frame in which the magnetic field was determined. • What if we switch frames to a frame where the velocity is zero?