Chapter 28 – Sources of Magnetic Field

1 Chapter 28 Sources of Magnetic Field - Magnetic Field of a Moving Charge- Magnetic Field of a Current Element- Magnetic Field of a Straight Current-Carrying Conductor- Force Between Parallel Conductors- Magnetic Field of a Circular Current Loop- Ampere s Law- Applications of Ampere s Law- Magnetic Materials1. Magnetic Field of a Moving Charge- A charge creates a Magnetic Field only when the charge is point:location of the moving point:point P where we want to find the Field . 20sin4rvqB = Magnetic Field from a point charge moving with constant speed 0= 4 10-7Wb/A m = N s2/C2= N/A2= T m/A (permeability of vacuum)c = (1/ 0 0)1/2 speed of lightMagnetic Field of a point charge moving with constant velocity20 4rrvqB = = vector from source to Field point rrr/ =Moving Charge: Magnetic Field Linesdirection of v.

2 Your fingers curl around the charge in direction of Magnetic Field The Magnetic Field lines are circles centered on the line of v and lying in planes perpendicular to that Direction of Field line: right hand rule for + charge point right thumb in 2. Magnetic Field of a Current Element- The total Magnetic Field caused by several moving charges is the vectorsum of the fields caused by the individual charges. Current Element: Vector Magnetic FieldnqAdldQ=20200sin4sin4sin42rIdlrAdlv qnrvdQdBdd === =20 4rrlIdB Law of Biot and Savart(total moving charge in volume element dl A)Moving charges in current element are equivalent to dQ moving with drift velocity.

3 (I = n q vdA)Current Element: Magnetic Field Lines - Field vectors (dB) and Magnetic Field lines of a current element (dl) are likethose generated by a + charge dQ moving in direction of Field lines are circles in planes to dl and centered on line of Magnetic Field of a Straight Current-Carrying Conductor22)sin(sin1 yxxdldldlrld+ = = = 2202/322024)(4axxaIyxdyxIBaa+=+ = xIaxaIB = = 24)2(00rIB = 20 =20 4rrlIdB If conductors length 2a >> xB direction: into the plane of the figure, perpendicular to x-y planeField near a long, straight current-carrying conductor- electric Field linesradiate outward from + line charge distribution.

4 They begin and end at electric Magnetic Field linesencircle the current that acts as their source. They form closed loops and never have end total Magnetic flux through any closed surface is zero there are noisolated Magnetic charges (or Magnetic monopoles) any Magnetic fieldline that enters a closed surface must also emerge from that Force Between Parallel Conductors- Two conductors with current in same direction. Each conductor lies in B set-up by the other generated by lower conductor at the position of upper conductor:rIB = 20 BLIF ='rILILBIF == 2''0- Parallel conductors carrying currents in same direction attract each other.

5 If I has contrary direction they repel each = 2'0 Two long parallel current-carrying conductorsForce on upper conductor is Forces and Defining the Ampere- One Ampere is the unvarying current that, if present in each of the two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2 x 10-7N per meter of Magnetic Field of a Circular Current Loop()2204axdlIdB+= 204rIdlB =()()2/1222204sinaxxaxdlIdBdBy++== =20 4rrlIdB ()()2/1222204cosaxaaxdlIdBdBx++== -Rotational symmetry about x axis no B component perpendicular to dl on opposite sides of loop, dBxare equal in magnitude and in same direction, dByhave same magnitude but opposite direction (cancel).

6 ()2/322202axIaBx+= ()()())2(4442/32202/32202/3220aaxaIdlaxa IaxadlIBx +=+=+= (on the axis of a circular loop)()2/322202axNIaBx+= Magnetic Field on the Axis of CoilaNIBx20 =2/3220)(2axBx+= (on the axis of N circular loops)(at the center, x=0, of N circular loops)(on the axis of any number of circular loops))(2aINAIN = =6. Ampere s Law- Law that allows us to obtain the magneticfields caused by highly symmetric currentdistributions. rIB 20= == IdlBldB0// Ampere s Law for a Long Straight ConductorIrrIdlBldB00)2(2 = == - Direction of current: right hand rule curl fingers of right hand around the integration path , the thumb indicates positive current + ++== addccbbadldlBdldlBdlBldB)0()()0(21// 00)(20)(2220110=+ + = rrIrrI)(2)(2202//101//cdarccircularrIBBa barccircularrIBB = = == arc = (angle) x (radius) = rFor an integration path that does not enclose the conductor.

7 B and dl antiparallel drBdlBldB = = cos IdIrdrIldB0002)(2 == = - This result does not depend on the shape of the path or on position of thewire inside If the path does not enclose the wire around integration path. =0 dAmpere s Law: General Statement- The total Magnetic Field at any point in the path is the vector sum of all fields produced by the individual conductors. -If the integration path does not enclose a wire IldB0 = 0 = ldB enclIldB0 = 0 = ldB does not mean that B = 0 everywhere along the path, only that Iencl= 0. 0 = ldB 7. Applications of Ampere s LawEx. Ex. Ex. 8. Magnetic Materials- An electron moving with speed v in a circular orbitof radius r has an angular momentum L andoppositely directed orbital Magnetic dipole moment.

8 It also has a spin angular momentum and oppositely directed spin Magnetic dipole = == 2/2revevrrrevAI = = = = 22)(22prL =emremrmvrprL 22= = = =Lme2= Model of electron in an atom- Atoms contain moving electrons, e- form microscopy current loops that produce Magnetic fields (randomly oriented, no net Bint). In some materials, external Bextcauses these loops to orient with Field , adding to the Bext magnetized momentum of e-: = = == -Atomic angular momentum is quantized: L ~h/2 (its component along a direction is always an integer multiple of this value). (h = Planck constant = x 10-34J s)- Associated with the quantization of L is an uncertainty in direction of L and of (since they are related).

9 Bohr Magneton- Electrons have intrinsic angular momentum(Spin) that is not related to orbital motion, but can be seen as spinning on an axis. The angular momentum has an associated Magnetic moment with magnitude B. Magnetic Materials- When Magnetic materials are present, the magnetization of the materialcauses an additional contribution to B. Paramagnetism- In an atom, most of the orbital and spin Magnetic moments add to zero. However, in case cases the atom has Magnetic moment B. If such atomis placed on B, the Field will exert a torque:B = this torque aligns the Magnetic moment with Magnetic Field (position of minimum potential energy).

10 In that position, the current loops add to the externally applied B produced by a current loop is proportional to loop s Magnetic dipole moment additional B produced by electron current loops proportional to totalper unit volume of material (V) = =Magnetization:- Additional Magnetic Field due to M of material is: 0 MUnits: (A m2)/m3= A/m - When a magnetized material surrounds a current-carrying conductor, the total B is:MBB 00 +=B0= Field caused by the current conductorbehavior typical of aparamagnetic material. - The Magnetic Field at any point in a paramagnetic material is greater by the factor Km(relative permeabilityof the material) than it will be if the material were replaced by vacuum.