Exercises with Magnetic Monopoles - Kurt Nalty

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Extending these equations to include magnetic monopoles, we have ⃗∇ · ⃗E = ρ e ɛ ⃗∇ · ⃗B = µρ m ⃗∇ × E ⃗ = − ∂ B ⃗ ∂t − µ ⃗j m ⃗∇ × ⃗ B = µɛ ∂ ⃗ E ∂t + µ ⃗j e Extended Lorentz Force The standard Lorentz force is ⃗F = q 0 ( ⃗E + ⃗v × ⃗ B ) Extending this to include monopoles, we have ( ( ) ⃗F = q 0 ⃗E + ⃗v × B ⃗ E + q m ⃗B − ⃗v × ⃗ ) c 2 Electric Field from a Point Charge The standard electric field due to a point charge q e at (x s , y s , z s ) seen at point (x, y, z) is ⃗E(x, y, z) = q e (x − x s )⃗a x + (y − y s )⃗a y + (z − z s )⃗a z 4πɛ ( (x − xs ) 2 + (y − y s ) 2 + (z − z s ) 2) 3/2 <strong>Magnetic</strong> Field from a Point Monopole The magnetic field due to a point charge q m (x, y, z) is at (x s , y s , z s ) seen at point ⃗B(x, y, z) = µq m 4π (x − x s )⃗a x + (y − y s )⃗a y + (z − z s )⃗a z ( (x − xs ) 2 + (y − y s ) 2 + (z − z s ) 2) 3/2 2
Field Angular Momentum due to Charge/Monopole Interaction This section follows homework problem 8, chapter 3 of Julian Schwinger’s Classical Electrodynamics. Place an electric charge at (0, 0, R/2). Place a magnetic charge at (0, 0, −R/2). The electric field is ⎡ ⎛ ⎤ ⃗E(x, y, z) = q e 4πɛ ⎞ ⎣−∇ ⃗ ⎝ 1 √ ⎠⎦ x 2 + y 2 + (z − R/2) 2 = q e x⃗a x + y⃗a y + (z − R/2)⃗a z 4πɛ ( x2 + y 2 + (z − R/2) 2) 3/2 The magnetic field is ⃗B(x, y, z) = µq m 4π = µq m 4π The power flux Poynting vector is ⃗S = 1 µ ( ⃗ E × ⃗ B) ⎡ ⎛ ⎞⎤ ⎣−∇ ⃗ ⎝ 1 √ ⎠⎦ x 2 + y 2 + (z + R/2) 2 x⃗a x + y⃗a y + (z + R/2)⃗a z ( x2 + y 2 + (z + R/2) 2) 3/2 = 1 q e x⃗a x + y⃗a y + (z − R/2)⃗a z µ 4πɛ ( x2 + y 2 + (z − R/2) 2) × µq m x⃗a x + y⃗a y + (z + R/2)⃗a z 3/2 4π ( x2 + y 2 + (z + R/2) 2) 3/2 = q eq m 1 yR⃗a x − xR⃗a y 16π 2 ɛ ( x2 + y 2 + (z − R/2) 2) 3/2 ( x2 + y 2 + (z + R/2) 2) 3/2 The momentum density is ɛµ ⃗ S = µq eq m 16π 2 yR⃗a x − xR⃗a y ( x2 + y 2 + (z − R/2) 2) 3/2 ( x2 + y 2 + (z + R/2) 2) 3/2 3
Page 1: Exercises with Magnetic Monopoles K
Page 5 and 6: ⃗∇(ρ e ρ m ) −1 . Each form
Page 7 and 8: We now write the field angular mome
Page 9 and 10: Looking at the radial component in
Page 11 and 12: ∫ ρ=∞ ρ=0 ρ 2 ∫ ρ=∞ (
Page 13 and 14: So, x = r cos θ sin φ y = r sin
Page 15 and 16: For the zone R/2 < r < ∞, we have
Page 17 and 18: Figure 1: tan −1 (4rR/(4r 2 − R
Page 19 and 20: if the ratio of magnetic to electri
Page 21 and 22: Electron Motion in a Radial Magneti
Page 23 and 24: τ = = = = [ µq e q m ⃗a × ⃗r
Page 25 and 26: ρ = r = mv 4πr2 µq e q m r = µq
Page 27 and 28: Without loss of generality, I can p
Page 29: [5] Alexander Russell, “The Magne

Exercises with Magnetic Monopoles - Kurt Nalty

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