Electric field near grounded conducting cylinder
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Consider a grounded conducting cylinder of radius, a, in a uniform electric field, E0 in the x direction. Let the cylinder axis be the z-axis. Using the Laplace's equation solution in cylindrical coordinates, find the potential and the induced surface charge.
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The expert examines the electric field near grounded conducting cylinder.
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In case of symmetry in the z-direction the solution of the Laplace equation is a superposition of terms Z_n of the form:
Z_n = (A_n r^n + B_n r^(-n))(C_n Cos(n theta) + D_n Sin(n theta)) for n >0
and in case of n = 0:
Z_0 = A_0 + B_0 Log(r)
Let's take the angle theta to be the angle w.r.t. the x-axis. Because the electric field is E_0 in the x-direction far away from the cylinder, you know that asymptotically the potential is:
V(r) = - E_0 r Cos(theta)
This in itself does satisfy the Laplace equation. You can get this term from Z_1 by taking A_1 = - ...
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