Vector field and Inductance. P.Ravindran, PHY041: Electricity & Magnetism 19 February 2013: Vector Field, Inductance.

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1 Vector field and Inductance

2 Earth s Magnetic Field Earth s field looks similar to what we d expect if 11.5 there were a giant bar magnet imbedded inside it, but the dipole axis of this magnet is offset from the axis of rotation by Also, the south pole of this magnet is near the geographic north pole, N G. A compass points in the direction of the magnetic north pole, N M, around which the field lines reenter Earth s surface. (Magnetic north is actually the south pole of Earth s magnetic dipole.) N M N G p g p ) μ orb N M, which is currently located in Greenland, drifts about over the centuries. About every million years Earth s field reverses entirely, as we know from the orientations of magnetic fields near the Mid Atlantic Ridge. The field is likely due to the motion of charged particles in the fluid outer core, and it protects us from an otherwise deadly dl solar wind.

3 Line ( Path) Integrals: Let v is a vector function dl is the infinitesimal displacement vector Integral carried out along a specified path from point a to point b gives line or path integral. b v.dl a Line integral depends critically on the particular path taken from a to b.

4 If path forms a closed loop i.e. a=b, the integral can be written as: vdl. There is a special type of vector which does not dependd on path, but is determined d entirely by the end points. The force which have this property called as conservative force.

5 Surface (Flux) Integrals: Let v is a vector function da is an infinitesimal patch of area (direction is to the surface) If v is flow of a fluid (mass per unit area per unit time) vda. S : total mass per unit time passing through the surface

6 If the surface is closed: cose. vda o For closed surface, direction of da is outward o For open surfaces it s arbitrary. osurface integral depends on the particular surface chosen. o There is a special class of vector function for which integral is independent of the surface, and is determined entirely by the boundary line.

7 Curl V k V j V V i k z j y i x z y x ˆ ˆ ˆ ˆ ˆ ˆ k j i ˆ ˆ ˆ z y x z y x V V V det How much a vector field causes something to twist z y x How much a vector field causes something to twist

8 Stokes Theorem This theorem will be used to derive Ampere s circuital law which is similar to Gauss s law in electrostatics. According to Stokes theorem, the circulation of A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L. L A dl S A ds The direction of integration around L is related to the direction of ds by the right-hand rule.

9 The fundamental theorem for Curls known as Stokes Theorem (Transformation Between Surface and Line Integrals) Let S be a piecewise smooth oriented surface in space and let the boundary of S be a piecewise smooth simple closed curve P.LetV(x, y, z) be a continuous vector function that has continuous first partial derivatives ina domain inspace containing S. Then S ( v ). da vdl. P o o Line integral It s general interpretation (not complete): The integral of a derivative over a region is equal to the value of the function at the boundary. Note: on the right hand side circle over integral indicates perimeter of the surface is closed.

10 ocomplete description: Derivative is the form of (curl) Region is (patch of surface) Boundary is the perimeter of the patch (due to perimeter here boundary indicates a closed line integral). Outcomes: ( v). da 1) S depends only on the boundary line, not on the particular surface used

11 Stokes Theorem Physical Meaning Considering a surface S having element da defined in the usual way and curve C denotes the curve describing the boundary of the surface S. If there is a vector field V, then the line integral of V taken round C is equal to the surface integral of V taken over S S V is the total circulation of V per unit area. Sumover all differential squares, vdl v.dl at all interior lines will cancel, only the contributions from the exterior lines remain. So the Stokes theorem the surface integral of the curl of a vector field over a three dimensional surface is equal to the line integral of the vector field over the boundary of the surface provides the transformation from a line integral to a surface integral in three dimensional space.

12 Significance of Stokes stheorems Stokes s Theorem relates a surface integral of a derivative construction of V (the curl) over a surface S to a line integral of V over S s boundary. V ds V d s S n S S S

13 Relation of the speed of light and electric and magnetic vacuum constants ε 0 permittivity of free space, also called the electric constant As/Vm or F/m (farad per meter) permeability of free space, also Vs/Am or H/m (henry per μ 0 called the magnetic constant meter)

14 Maxwell s equations: integral form Gauss's law Gauss's law for magnetism: no magnetic monopole! Ampère's law (with Maxwell's addition) Faraday's law of induction (Maxwell Faraday equation)

15 Differential operators the divergence operator div t the curl operator curl, rot A A x y A A x y z xˆ yˆ zˆ A x y z x A A A x x y z the partial derivative with respect to time z Other notation used

16 Transition from integral to differential form Gauss theorem for a vector field F(r) Volume V, surrounded by surface S Stokes' theorem for a vector field F(r) Surface, surrounded by contour

17 Maxwell s equations (SI units) differential form density of charges j density of current

18 Electric and magnetic fields and units E electric field, volt per meter, V/m B the magnetic field or magnetic induction tesla, T D electric displacement field coulombs per square meter, C/m^2 H magnetic field ampere per meter, A/m

19 Torque & Force on Magnetic Dipoles In a uniform magnetic field; net force is zero. F I dl B ( I dl ) B 0 While in non uniform magnetic field, it is non zero and can be expressed as F ( m. B) ) Similar to the electrostatics case.

20 df i ds B 2 1 Net force will be downward on the loop. U B U 2 2 1z zb

21 U zb z du F 21 dz d zb 21 dz 2 z F 1 F 21 2 z db dz 1 z Proved that force will be zero in a uniform magnetic field.

22 Induced d magnetic dipole moments

23 Induced magnetic dipole moments An applied magnetic field can induce dipole moments I I ind I+I ind Downward Induced magnetic dipole moment.

24 Important points In a non uniform magnetic field, permanent dipoles are attracted towards the source of the field Induced dipoles are repelled from the source of the field. Similar induced moment effect is observed in materials that lack permanent magnetic dipole moments.

25 Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which h the magnetic flux through the circuit is changing with time. d B N d B dt dt If coil has N turns

26 Magnetic Flux is defined as B B d A Change in flux may be due to Change in magnetic field Change in the area Both.

27 Lenz slaw Law The induced current in a circuit will flow in such a direction that the produced flux tends to oppose the change in the flux B that causes the induced current.

28 Lenz s Law B, H N S I induced V+, V Lenz s Law : emf appears and current flows that creates a magnetic field that opposes the change. Hence the negative sign in Faraday s law.

29 Lenz s law The flux of the magnetic field due to the induced current opposes the change in the flux that causes the induced current. d B dt

30 Induced current flows in the loop Et External agent pulls the loop with constant nt speed

31 B BA BDx B d B dt BDv I ind R BDv R

32 F is the net magnetic force 1 If external agent pulls with constant speed F ext = F 1 = I ind DB ext 1 ind Mechanical power P = F 1 v

33 The power expended by the external agent P F v 1 P I ind DBv P D 2 B R 2 v 2

34 A conducting rod of length L is being pulled along horizontal, frictionless and conducting rails. A uniform magnetic field fills the region in which the rod moves. Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s, resistance of rod as 415 m.

35 Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s resistance of rod as 415 m Find Induced emf = BLv = V Current in the conducting loop. I = /R / = 1.49 A

36 Force that must be applied by external agent to maintain its motion F = ILB = N At what rate does this force do work on rod? P = F v = W

37 2. Gauss law for Magnetic Fields Gauss law for magnetic fields is a formal way of saying that magnetic monopoles do not exist. The law states that the net magnetic flux B through any closed surface must be zero. Gauss law for magnetic fields ** says there can not be a net magnetic flux through the surface since there can be no net magnetic charge enclosed by the surface. ** Maxwell s second equation B ds B 0 For comparison, Gauss law for electric fields E EdS Q enc 0 In both equations, the integral is taken over a closed Gaussian surface. Gauss law for electric fields says this integral is proportional to the net electric charge Q enc enclosed by the surface.

MAGNETIC FLUX It is defined as the magnetic lines of force produced in the medium

magnetic flux density. B ds s weber. B The unit of magnetic flux is T.m 2 ( weber).

small area or B is also defined as B = H where, H =magnetic field (A/m) = permeability of

38 MAGNETIC FLUX It is defined as the magnetic lines of force produced in the medium surrounding electric currents or magnets and is expressed as surface integral of the magnetic flux density. B ds s weber. B The unit of magnetic flux is T.m 2 ( weber). And B is defined as magnetic flux per unit area (Magnetic flux density) through a loop of small area or B is also defined as B = H where, H =magnetic field (A/m) = permeability of the medium (H/m) = 0 r = = -7 0 permeability of free space 4 x 10 H/m r = relative permeability of the medium 0

Chapter 30. Induction and Inductance

Chapter 30. Induction and Inductance Chapter 30 Induction and Inductance 30.2: First Experiment: 1. A current appears only if there is relative motion between the loop and the magnet (one must move relative to the other); the current disappears