This test has 5 questions to be completed in 80 minutes. The value of each quest

Question

This test has 5 questions to be completed in 80 minutes. The value of each question is 10 points, to a maximum of 50 points. Read the questions carefully, for some of them are easier than others. Please show your results and work in this questionnaire. No work, no credit. I will provide you with extra paper for calculations. An infinite conducting cone, with opening angle 45 degree, lies with its axis perpendicular to an infinite conducting plane, and with its tip nearly touching, but insulated from, this plane. The cone is held at potential V_0, and the plane is grounded (V = 0). Write down Laplace's equation in the coordinate system appropriate to this geometry, and indicate which two of the terms drop out because of the independence of the potential on the corresponding coordinate. Solve the resulting equation for the electric potential, subject to the boundary conditions, and compute the electric field vector from the potential.

Explanation / Answer

First of all Laplace's equation combines with Poisson's equation transforms the Gauss' law in the following equation for a linear medium,

2V= [ V]/

Now solutions of this equation are given in Cartesian, Spherical and cylindrical coordinates using the Laplace operator 2 . Now according to uniqueness theorem, the solutions must satisfy some boundary conditions for V. As an example Vd =0 at the boundary as well as V1 = V2 at the boundary. Now for infinite conducting cone we use Taylor's conical model which shows that small electric volume in an electric field deforms its shape due to surface tension and create a cone shape of convex sides. But Taylor 's model of cone changes the Laplace equation structure for only field distribution of the electrostatically stressed cone. Therefore the field distribution condition is given by,

V[ R, ] = V0 + A * R 1/2 * P1 /2 cos .............. {1}

the field production condition is

R = R0 [ P1 /2 cos ]

But this condition does not satisfy the Laplace equation for an infinite rigid cone model. Considering the bracket part as non-zero and field distribution in apex and sides differ highly we get the changed form of Laplace solution for this rigid infinite cone. The equation is:

A) 2V = 1/R2 * / R [R2 * V / R ] + 1/R2 Sin * / [Sin * V/ ] =0 .....................{3}

We choose the spherical coordinate system for this cone problem as = 0 in the surface contact of the cone with the conducting plane.

B) The general solution for this problem becomes,

V[ R, ] =   ( Av Rv + BvR-v-1)* Pv Cos

Now considering Laplace conditions in the inside and near the opening angle of the cone while taking approximation for the outside contact with conducting plane as well as using the expansion coefficient we get the changed solution,

V[ R, ] = V0 + ( AvsRvs)* Pvs Cos

Again we consider the insulating layer between the two infinite conducting surfaces and that means the exact solution has no dependence on R

Therefore the exact solution is

V' (R' ') = V'(')= V0 [ln Tan '/2]/ln (Tan 0/2)

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