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Sources of the Magnetic bb Field Lesson 8 Biot-Savart Law Force between parallel conductors Ampere’s Law Use of Ampere’s Law E Field Interaction between stationary charges is mediated by an Electric Field E Field B Field Interaction between currents is mediated by a Magnetic Field B Field Electric Dipoles align along an E Field E - + E Dipole Magnetic Dipoles align along a B Field B S N B Dipole Biot-Savart Law Biot-Savart Law For a point source charge dq the E and B Fields due sources electric field produced at a position r from the source is 1 dq r 1 dq ˆ dE= r = 2 3 4pe0 r 4pe0 r For a length of wire ds with current I (this is the same as charge per unit length dq with velocity v ) the magnetic field produced at a position r from the source d s is I m0 I m 0 dB = ds ˆr = ds r 2 3 4p r 4p r B Field due to source m 0 = 4p 10 - 7 Tm A = Permeability B Field due to loop Right Hand Rules Right Hand Rule I Need B and v then get FB Right Hand Rule II Need I get B Magnetic Fields from conductors of different shapes k B Field from wire Magnitude of Magnetic Field about thin conductor B(r ) = B(r ) = R q ds r m0I 2pr m0I B(r) = d B = 2 sin q ds k r p 2 - 2 2 r = s + R sin q = sin ( p - q ) = 2 2 ; r s +r m 0 I r = B (r ) 3 ds k 2 2 p 2 2 +r ) - (s B Field from closed wire segment I r q X m0I q B= 4 pr B Field from closed loop I B B= m0 I 2r Picture I B k Two parallel conductors Parallel wires I j i B2 I1 a I2 L B1 Force on wire F1 = 1 due to field of wire 2 I1 L B 2 = I1L j B 2 B2 F1 = - m 0 m 0 I2 = i 2p a I1 I2 L m 0 I1 I2 L j i = + k 2pa 2pa Force on wire 2 due to field of wire 1 I2 L B 1 = - I2 L j B 1 F2 = B1 = - F2 = m 0 I1 I2 L 2pa m 0 I2 2 pa j i = - i m0 I1 I2 L 2pa k k Two parallel conductors j i Parallel wires II B2 F1 a F2 B1 Parallel wire rules Parallel Currents Attract Anti Parallel Currents Repel a = 1m I1 = I2 = I = 1A Definition of Amp m F= 0 = 2 10 N -7 m 2p This defines the Amp . as: 2 10 N m measured between wires one meter apart This in turn defines the Coulomb as : The quantity of charge that flows through any cross section of a conductor in one second when a steady current of one amp is flowing . The current flowing when a force of -7 Recall Gauss ' s Law Gauss's Law for E Q= d A = e E EA Gaussian surface as E = constant and E || d A everywhere on surface For magnetism B dA = 0 as magnetic field has ALWAYS been observed to be produced by magnetic dipoles In electrostatics, potential difference is defined by for B Gauss's Law where integral is evaluated along ANY path b V ab = E ds a starting at a and finishing at b, thus E ds = 0 as E is a conservative force per unit charge For magnetism, in general B ds 0 and B ds = Amperian Loop B ds Amperian Loop : B = constant and B ds Amperes B d s Law = B ( RI ) I B R Amperian Loop ds Amperian Loop for steady current m 0 = I (2 p R ) = m 0 I 2 p R true for ANY closed path Amperes Law Amperes Law II B ds = m 0 ITotal ITotal = sum of all currents threading loop Equivalence of Laws Gauss’s Law is Equivalent to Coulombs Law Biot-Savart Law is Equivalent to Amperes Law. Toroids and Solenoids I I Toroid Solenoid Toroids and Solenoids II B B ds = B ds = B loop 2 path 1 w path 1 B d s = m 0 NI = BL loop 1 3 l ds = BL N B = m0 L 4 Amperian Loop I= m 0 nI Picture One can perform the same Amperian calculation the Toroid III Toroids and for Solenoids and get the same result B = m 0 nI where n is the number of turns per length N n= but now 2pR , R = Radius of Toroid instead of N n= L , for Solenoid Galvanometer Galvanometer