BII 1 Currents make B fields K We have seen that charges make E fields dE 1 dQ r 4 0 r 2 K K 0 I d A r Currents make B fields according to the Biot Savart Law dB 4 r2 where 0 constant 4 10 7 SI units dB is the element of B field due to the I element of current I dl dl is an infinitesimal dl length of the wire with direction given by r r the current The total B field due to the entire K K K 0 I d A r current is B dB 4 r2 dB I This can be a very messy integral This law was discovered experimentally by two French scientists Biot and Savart in 1820 but it can be derived from Maxwell s equations Example of Biot Savart B field at the center of a circular loop of current I with radius R Here the integral turns out to be easy B K I dA d A r d A r sin 900 1 dB 0 4 R2 We can replace the vector integral K K B dB with the scalar integral B R dB r dl I because all of the dB s point in the same direction B 0 I dB 4 R N dA 2 2 R 0 I 2R B The full field at all positions near a current loop requires very messy integrations which are usually done numerically on a computer The full field looks like this Last update 10 21 2009 I Dubson Phys1120 Notes University of Colorado BII 2 Another more difficult example of the Biot Savart Law B field due to a long straight wire with current I The result of a messy integration is r I B r dB r dl 0 I 2 r This formula can be derived from a fundamental law called Ampere s Law which we describe below The B field lines form circular loops around the wire B To get directions right for both these examples B due to wire loop B due to straight wire use Right hand rule II With right hand curl fingers along the curly thing your thumb points in direction of the straight thing I Force between two current carrying wires Current carrying wires exert magnetic forces on each other Wire2 creates a B field at position of wire 1 Wire1 feels a K K K force due to the B field from wire 2 Fon1 I1 L B2 I2 I1 Bfrom I2 from 2 Fon 1 from 2 I I1 L B2 I1 L 0 2 2 r force per length between wires F F F I I 0 1 2 L 2 r Parallel currents attract Anti parallel currents repel r I1 I2 Going my way Let s go together Going the other way Forget you L F F I1 F I2 F r Last update 10 21 2009 Dubson Phys1120 Notes University of Colorado BII 3 Gauss s Law for B fields B field lines are fundamentally different from E field lines in this way E field lines begin and end on charges or go to But B field lines always form closed loops with no beginning or end A hypothetical particle which creates B field lines in the way a electric charge creates E field lines is called a magnetic monopole As far as we can tell magnetic monopoles magnetic charges do not exist There is a fundamental law of physics which states that magnetic monopoles do not exist Recall the electric flux through a surface S is defined as E K K E da In the same S way we define the magnetic flux through a surface as B K K B da Gauss s law S stated that for any closed surface the electric flux is proportional to the enclosed electic charge B E B M Yes Impossible OK K K q E v da enc0 But there is no such thing as magnetic charge so the corresponding equation for magnetic fields is K K B v da 0 This equation which has no standard name is one of the four Maxwell Equations It is sometimes called Gauss s Law for B fields Ampere s Law Ampere s Law gives the relation between current and B fields For any closed loop L K K v B d A 0 Ienclosed where Ienclosed is the current L through the loop L Last update 10 21 2009 Dubson Phys1120 Notes University of Colorado BII 4 It will turn out the Ampere s Law is only true for constant current I If the current I is changing in time Ampere s Law requires modification Ampere s Law for steady currents like Gauss s Law is a fundamental law of physics It can be shown to be equivalent to Biot Savart Law We can use Ampere s Law to derive the B field of a long straight wire with current I B field of a long straight wire L imaginary circular loop of radius r We know that B must be tangential to this loop B is r purely azimuthal B can have no radial component toward or away from the wire How do we know loop L I this A radial component of B is forbidden by Gauss s Law for B fields Alternatively we know K K from Biot Savart that B is azimuthal So in this case B d A B dA Also by symmetry the magnitude of B can only depend on r distance from the wire B B r K K B v d A L K K B d A B v B d A Bconst B v d A B 2 r 0 I 0 I 2 r Like Gauss s Law Ampere s Law is always true but it is only useful for computing B if the situation has very high symmetry B field due to a solenoid solenoid cylindrical coil of wire It is possible to make a uniform constant B field with a solenoid In the limit that the solenoid is very long the B field inside is uniform and the B field outside is virtually zero Consider a solenoid with N turns length L and n N L turns meter Last update 10 21 2009 Dubson Phys1120 Notes University of Colorado BII 5 L B I End View I I B uniform inside Side View I out B I in B 0 outside loop L l K K B nA I v d A B A 0 I thru L 0 N L B field inside solenoid is B 0 n I turns thru L B 0 n I 0 N I L Permanent Magnets Currents make B fields So where s the current in a permanent magnet like a compass needle An atom consists of an electron orbiting the nucleus The electron is a moving charge forming a tiny current loop an atomic current In most metals the atomic currents of different atoms have random orientations so there is no net current no B field In ferromagnetic materials Fe Ni Cr some alloys …
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