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3.2.2.     Magnetic Field Sources <CJ chap 21.7-21.10  >

3.2.2.1. Discussion

3.2.2.1.1.     The Right Hand Rule (RHR): determines the x product, and the direction of B around currents

3.2.2.1.2.     Magnetism in matter arises from the currents in matter and magnetic moments of particles

3.2.2.1.2.1. The magnetic moment of a loop of current is the ‘fundamental magnet’

3.2.2.1.2.2. Use the RHR to get the N and S poles for such a loop.

3.2.2.1.3.      

3.2.2.2. Mathematical

3.2.2.2.1.     Biot-Savart law: Magnetic fields arise from the motion of electric charge, i.e. electric currents 

3.2.2.2.1.1. DB = (m_o/4p) I Ds x r_unit / r² where I = current, Ds = length of wire, DB = mag. Field

3.2.2.2.1.2. (m_o/4p) = k_m = 1E-7 exactly thus defining the value of m_o, the permeability of free space

3.2.2.2.1.3. The unit vector r_unit points from the current segment Ds to the point r where B is to be found

3.2.2.2.2.     B = m_o I /(2pa) gives the magnetic field a distance ‘a’ from an infinite straight wire

3.2.2.2.3.     B = m_o I R² /(2 x² + R²)^3/2 = B field on the axis a distance x from a circular loop of current I, Radius R,

3.2.2.2.4.     F/s = m_o I₁ I₂ /(2pa) = force between two long parallel wires a distance ‘a’ apart with currents I₁  and I₂

3.2.2.2.4.1. Defines the Ampere if the force per m = 2E-7 results from equal currents I₁  and I₂ of both 1 Amp

3.2.2.2.5.     Ampere’s law:  B x distance around a closed loop = m_oI

3.2.2.2.6.       B = m_o n I  = B field in a solenoid with n = N / l   (# of turns per length)

3.2.2.3. Advanced

3.2.2.3.1.     Biot-Savart law:  dB = (m_o/4p) I ds x r_unit / r² where I = current, ds = length of wire, DB = mag. Field

3.2.2.3.2.     Gauss Law for Magnetism     B  ds  = 0  = the magnetic flux through any closed surface  

3.2.2.3.3.     Ampere’s law:     B  ds  = m_o I

3.2.2.3.4.     Ampere’s law modified by Maxwell displacement current  B  ds  = m_o I  + m_o e_o d( B  ds)/dt

3.2.2.3.4.1. Using a cylindrical surface around a wire ending in a capacitor then EA = Q/e_o

3.2.2.3.4.2. thus e_o dF/dt = dQ/dt  = I_Maxwell  & using this I_Maxwell in addition to the I in Amperes law gives result

3.2.2.3.5.     The Magnetization vector, M, = magnetic moment per unit volume and

3.2.2.3.5.1. Thus B = B₀ + B_m     =    B₀ + m_oM   =  m_o (H + M) 

3.2.2.3.5.2. For paramagnetic and diamagnetic substances,  M = c H  where c = the magnetic susceptibility

3.2.2.3.5.2.1.     with  m_m =  m_o (1 + c) substances are classified as

3.2.2.3.5.2.2.     paramagnetic  m_m > m₀ , diamagnetic m_m < m₀ ,  and ferromagnetic  m_m >> m₀