Chapter 5 Summary 5.1 Introduction and Definitions

Transcription

1 Chapter 5 Summary 5.1 Introduction and Definitions Definition of Magnetic Flux Density B To find the magnetic flux density B at x, place a small magnetic dipole µ at x and measure the torque on it: N = µ B(x) (5.1) The torque tries to turn the dipole in the direction of B. Conservation of charge condition: ρ=charge density (coul/m 3 ) J=current density (amps/m 2 ) ρ t + J=0 (5.2) PHYS 532. L6 1

2 5.2 Biot-Savart Law Force on a volume V: F = V Torque on volume V: N = V J B d 3 x x (J B) d 3 x Magnetic force on a point charge q moving at v: F = qv B (5.12) (5.13) PHYS 532. L6 2

3 5.3 Differential Equations of Magnetostatics and Ampere s Law Integral form of Biot-Savart Law: B(x) = µ o 4π x x' J(x' ) x x' 3 where µ o =4π 10-7 d 3 x'= µ o 4π J(x') d 3 x' (5.14) x x' (5.16) Differential equations: Equation (5.16) obviously implies B=0 For static conditions ( J=0), (5.16) implies B = µ o J (5.17) (5.22) Integrating (5.22) over a closed area gives Ampere s law: C B dl = µ o I where I is the current threading the closed loop C. (5.25) PHYS 532. L6 3

4 5.4 Vector Potential Since B=0, any magnetic flux density can be represented in terms of a vector potential: B = A (5.27) Integral expression for A: A(x) = µ o J(x') 4π d 3 x' + Ψ(x) (5.28) x-x' PHYS 532. L6 4

5 5.6 Magnetic Fields of a Localized Current Distribution In (5.28), use an expansion like the one we used to get the multipole expansion in electrostatics: 1 x x' = 1 x + x x' x First term gives zero (no magnetic monopole). Second term gives A(x) = µ o 4π where m is the magnetic dipole moment: (5.28) m x x 3 (5.55) m = 1 x' 2 J(x')d 3 x' (5.54) If the current is a plane loop, the magnetic moment has magnitude equal to the current in the loop times the area of the loop. B(x) = µ o 4π Looks like E of electric dipole 3x(x m) m x 3 + 8π 3 mδ (3) (x) Extra term to make integrals right including origin PHYS 532. L6 5 (5.64)

6 5.7 Force and Torque on and Energy of a Localized Current Distribution in an External Magnetic Induction Force on a dipole in an external magnetic induction B: F = (m B) (5.69) Dipole aligned with B is drawn toward the region of strong field. N = m B (5.71) PHYS 532. L6 6

7 5.8 Macroscopic Equations, Boundary Conditions on B and H For materials, define M = magnetic moment per unit volume The magnetic effect of that continuum of dipoles is equivalent to a current distribution (called magnetization current) J M = M (5.79) PHYS 532. L6 7

8 Magnetic Maxwell equations in a medium: The flux-conservation Maxwell equation, which is homogeneous, remains the same as in a vacuum: B=0 (5.75) For the Ampere s Law Maxwell equation, split the current into two parts: magnetization current and current carried by free particles: B = µ o ( J + M) (5.80) Define the magnetic field H: Current carried by free charges H = 1 µ o B M (5.81) The magnetic field Maxwell equations in a medium then become B=0 H = J (5.82) For a linear, isotropic medium, it is convenient to write B = µh (5.84) where µ is called the magnetic permeability. For ferromagnetic materials, B is a nonlinear function of H. PHYS 532. L6 8

9 Note that H= M If M is specified (as is typically the case for problems involving ferromagnetic materials), then H = 0 and the equations of magnetostatics are equivalent to the equations of electrostatics. Among working scientists (as opposed to textbook authors), there is often confusion about H and B. Plasma physicists usually use the symbol B and avoid H, but they refer to B as the magnetic field, which is politically incorrect. Boundary conditions: The normal component of B is conserved at the boundary between magnetic materials. The tangential component of H is conserved at such a boundary if the only current flowing on the boundary is magnetization current. PHYS 532. L6 9

10 5.15 Faraday s Law of Induction Integral form: where E = k df dt PHYS 532. L6 10 (5.135) the EMF in the circuit is defined by E = E' dl (5.134) C and E is the electric field in the rest frame of the curve C, which may be moving. The magnetic flux threading the circuit C is defined by F = B n ˆ da (5.133) where S = surface bounded by C. S In SI units, the constant k is 1. For gaussian units, it is 1/c. Differential equation form of Faraday s Law (SI units): E + B t = 0 (5.134) Faraday s Law was an experimental discovery. However, it could almost have been derived from consideration of the properties of E and B under Galilean transformation. E' = E + v B B' = B (5.142)

11 Lenz s Law: Intuitive way to see the sign of the induced electric field: The induced current (and accompanying magnetic flux) is in such a direction as to oppose the change of flux through the circuit: Suppose B is out of page and increasing with time. Induced E is clockwise. In a conducting wire, it would drive a clockwise current, which, by Biot-Savart law, would cause a magnetic field into the page. PHYS 532. L6 11

12 5.16 Energy in the Magnetic Field Magnetic fields by themselves don t change particle energies, because the magnetic force is perpendicular to the particle velocity. Therefore, it is not possible to discuss magnetic energy in the context of magnetostatics. However, because time-dependent magnetic fields imply electric fields, creation of a magnetic field configuration requires energy. Expressions for change in energy associated with change in magnetic field: δw = δa J d 3 x = δb H d 3 x (5.144, 5.147) In a linear medium, with B proportional to H, W = 1 2 B H d3 x = 1 2 J A d3 x (5.148, 5.149) PHYS 532. L6 12