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2d Coulomb and Biot-Savart laws

  • Thread starter Tsunami
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I'm trying to find some sort of simple derivation of these laws in 2d, using the integral expressions of the Maxwell equations.

For 2d Coulomb, I found this:

Imagine a rod of infinite length along the z-axis, carrying a charge q which is uniformly divided:

rho (the charge/volume) = lambda*dz*delta(x)*delta(y)

with dq= lambda*dz

So, using the electrical Gauss' law, Er being the value of E in radial direction:

eps0*Er*2*pi*r*dz = lambda*dz

Er= lambda/(eps0*2*pi*r)

Er =-dV/dr

=> V= -lambda/(2*eps0)*ln(r)

So, Coulomb's law would be, with pi(r´) being the electrical charge of the surface evaluated in point r´, en ds´ being the surface that is integrated:

V(r) = -lambda/(2*eps0)* int ( ln(r-r´) , ds´) ?

Does that make any sense? Or is this way off the mark?

And for the Biot-Savart law in the same manner, will the vector potential be something analogous to this thing?

Thank you,

W.
 

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