Lecture 22 Chapter 30 Magnetic Fields Due to Currents Review Used Biot Savart law to calculate B field from a loop at a point along the z axis B z 0iR 2 2 R 2 z 2 3 2 At point z R rewrite r r 0 B z 3 2 z Current carrying coil acts as a magnetic dipole Experiences a torque in an external B field Generates its own intrinsic B field Review Current carrying wires will exert a force on one another Calculate B field from wire a at site of wire b 0ia Ba 2 d Force on b from a is Fba 0 Li a ib ib LB a 2 d Parallel currents attract anti parallel currents repel B Fields from Currents 50 For certain symmetric distributions of charge able to use Gauss law to calculate E field r r q enc E dA 0 Integrate around closed loop For symmetric distributions called Amperian of charge use Ampere s loop law to calculate B field r r B d s 0 i enc B Fields from Currents 51 Use the right hand rule to determine the signs for the currents encircled by the Amperian loop Curl right hand around Amperian loop with fingers pointing in direction of integration Current going through loop in general opposite direction of thumb is positive negative B Fields from Currents 52 Use Ampere s law to calculate B field from long straight wire r r B d s 0ienc At every point of the Draw Amperian loop as a circle surrounding loop the wire Magnitude of B is constant Remind you of the magnetic field lines B and ds are tangent B Fields from Currents 53 B and ds are so cos cos0 1 r r B ds Bds B constant on loop so r r B d s B ds ds 2 r r r B d s B 2 r B Fields from Currents 54 Ampere s law becomes B 2 r 0ienc Current enclosed is just i so 0i B 2 r Same result as with Biot Savart law B Fields from Currents 55 Calculate B field inside a long straight wire r r B d s i 0 enc Again B and ds are and B is a constant so r r B ds B ds B 2 r B 2 r 0ienc B Fields from Currents 56 Need to find ienc Current is uniformly distributed so i enclosed by loop is to area enclosed r i 2 R 2 ienc r B 2 r 0i 2 R 2 0i B r 2 2 R B Fields from Currents 57 What happens if there are several loops of wire put together A long tightly wound helical coil of wire is called a solenoid Bend solenoid so ends meet to make a hollow donut gives a toroid Use Ampere s law to calculate B field for a solenoid and a toroid B Fields from Currents 58 Solenoid s B field is vector sum of fields produced by each turn loop in solenoid Near loop acts as infinite straight wire Between the loops An ideal solenoid fields tend to cancel is infinity long with closely Inside the solenoid packed turns of wire far from the wire B has uniform B field which is parallel to solenoid axis field is parallel to axis B Fields from Currents 59 For points outside the solenoid B fields from the upper parts of the turns tend to cancel the lower Ideal solenoid Boutside 0 For a real solenoid can Use right hand rule to assume Boutside 0 if find direction of B field length diameter Only consider points not near ends of solenoid Grasp solenoid so fingers follow direction of i in loops thumb points in B B Fields from Currents 60 Use Ampere s law to calculate B field of ideal solenoid r r B d s 0ienc Draw Amperian loop abcda intersecting solenoid Integral can be written as sum of 4 integrals one for each side r r br r cr r B ds B ds B ds a b r r ar r B ds B ds d c d B Fields from Currents 61 First integral B field is to ds b r r b B d s B s a Bh a For sides bc and da B is to ds so For the length outside the solenoid B 0 d r c r B ds 0 c r r B ds b a r r B ds 0 d r r B d s Bh B Fields from Currents 62 Now need to find amount of current enclosed r r B d s 0 i enc Single coil has current i But Amperian loop encloses several coils so total current is n is the number of turns per unit length i enc inh N n L N total of turns L length B Fields from Currents 63 Substituting into Ampere s law r r B d s 0 i enc Bh inh For ideal solenoid B 0in n is turns length B field of solenoid does not depend on diameter or length of solenoid is uniform over its cross section B Fields from Currents 64 Calculate B field for a toroid using Ampere s law r r B d s 0 i enc Choose Amperian loop to be a concentric circle inside toroid B and ds are parallel along entire loop so r r B ds B ds B 2 r B Fields from Currents 65 Current enclosed by loop is i enc iN r r B d s 0 i enc B 2 r 0iN B field for toroid is 0iN B 2 r B Fields from Currents 66 Toroid B field is not constant over its cross section 0iN 1 B 2 r N total of turns Use right hand rule to find direction of B field Grasp toroid with fingers in direction of current in windings thumb points in B B 0 outside toroid B Fields from Currents 67 Solenoids are practical way to setup a known uniform B field Like parallel plate capacitor to generate known uniform E field Many everyday devices use solenoids Example Tevatron at Fermilab B 0in B Fields from Currents 68 Tevatron is the largest of 6 synchrotons at Fermilab Accelerates protons and anti protons up to 1 TeV 1 TeV 1012 eV Remember a synchroton accelerates charged particles in a circular path of fixed radius by varying the B field mv r qB Chicago Wrigley Field Booster CDF Tevatron p source Main Injector new D B Fields from Currents 69 Tevatron uses 1000 magnets with B fields of 4 2 Tesla Small bar magnet 10 2 T Earth is 3x10 4 T Magnets are solenoids Niobium titanium alloy N 11 million Current 4000 A B Fields from Currents 70 Collider Detector at Fermilab CDF also uses solenoid Measures momentum and charge of particles by their path in a B field B Fields from Currents 71 CDF solenoid Niobium titanium copper and Al Length 5 m Diameter 3 m N 1164 Current 5000 A N B 0in 0i …
View Full Document
Unlocking...