By Norbury J.W.
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63) 40 CHAPTER 2. VECTORS Let dlr and dlθ be infinitessimal increments of length obtained when moving in the eˆr and eˆθ directions respectively. From Fig. 65) and as shown in Fig. 11. 66) The circumference of a circle (for which plane polar coordinates are eminently appropriate) is given by dlθ = 2π 0 rdθ = 2πr. The infinitessimal area element (magnitude only) is dA ≡ dlr dlθ = rdrdθ so that the area of a circle is πR2 . 67) R 2π 0 rdr 0 dθ = ( 12 R2 )(2π) = Spherical (3-dimensional) Polar Coordinates This coordinate system is specified by (r, θ, φ) and unit vectors (ˆ er , eˆθ , eˆφ ) shown in Fig.
Let dlx and dly be infinitessimal increments of length obtained when moving in the ˆi and ˆj directions respectively. 47) and The infinitessimal displacement vector is then dl = dlxˆi + dly ˆj = dxˆi + dyˆj. 48) The infinitessimal area element (magnitude only) is dA ≡ dlx dly = dxdy. 49) in plane rectangular coordinates. Thus the area of a rectangle (for which plane rectangular coordinates are eminently appropriate) is given by Area W of Rectangle = dA = L 0 dx 0 dy = LW . That is, the area of a rectangle equals length times width.
MAXWELL’S EQUATIONS Gauss’ law, Faraday’s law, Amp`ere’s law etc. However it was James Clerk Maxwell, who in 1861, unified all the previous work, made some significant corrections of his own, and finally wrote down what are today called Maxwell’s equations. 4) ×E+g and Amp`ere’s law ×B− dq ) and j is the where ρ is the charge density (charge per unit volume ρ ≡ dτ di ˆ current density (charge per unit area j ≡ dA A). k and g are constants and c is the speed of light in vacuum. 5) which gives the force F on a particle of charge q moving with velocity v in an electromagnetic field.
Classical electrodynamics for undergraduates by Norbury J.W.