Which are the stations in Circle Line?

Show that a semi-infinite line charge is the equivalent of a quarter circle line charge?

• A semi-infinite line charge of charge density σ (charge/meter) has begins at the origin, i.e., x = 0--> ∞. A test charge is located a at the end of the line charge and a perpendicular distance r from the line charge. Show that with respect to the test particle, the semi-infinite line charge is the equivalent of a quarter circle of radius r line charge. Here is a picture: http://i278.photobucket.com/albums/kk114/Remo_Aviron/elect2.jpg

• Answer:

  They are equal. I'll come back with proof it time permits. Edit: For the quarter circle: dQ=σrdθ k = 1/(4πε) dE = kdQ/r^2 r_hat = -cosθi + sinθj Ex = ∫kdQ/r^2i . r_hat [0,π/2] Ex = σ /(4πε*r) ∫i. -cosθi dθ [0,π/2] = -kσ/r Ey = ∫kdQ/r^2j . r_hat Ey = σ /(4πε*r) ∫ j.sinθj dθ [0,π/2] = kσ/r For the semi infinite line charge: dQ=σdx dE = k(σdx)/r^2 r_hat = xi + rj / √x^2+r^2 Ex = ∫kdQ/r^2i . r_hat Ex = ∫kσdx/r^2i . r_hat=kσ ∫x/(x^2+r^2)^(3/2) [0, ∞] Ex = kσ*-1/√r^2 + x^2) |0, ∞ limEx x-->0 = -kσ/r limEx x-->∞ = 0 Ey = ∫kdQ/r^2j . r_hat Ey = ∫kσdx/r^2j . r_hat=kσ∫ rdx/(x^2+r^2)^(3/2) [0, ∞] Ey = kσr*x/(r^2*√r^2 + x^2)|0, ∞ limEy x-->0 = 0 limEy x--> ∞ = kσ/r