Unable to Evaluate Integral
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Hello, I am actually trying to reproduce Arnold Sommerfeld's 1909 problem of a vertical electric dipole above the surface of the earth. While I tried extensively, to evaluate the integral in the field expression, my code always returns the expression as it is without solving for the integral ! Please help me out on this:
This is my code:
% An Attempt to Solve the Sommerfeld Problem of a Vertical Electric Dipole
% at a Height "h" Above the Surface of the Earth.
% Pre-defining Various Constants to be Used in the Problem:
epsilon0 = (8.85418782)*(10^(-12));
mu0 = (1.25663706)*(10^(-6));
epsilon_r = 8;
sigma = 0.1;
Il = 1;
h = 1;
% Input Variables:
f = input('Enter the value of frequency(in MHz) = ');
% Formulas to be Used in the Problem:
omega = (2*pi*(f*(10^6)));
k_0 = omega*(sqrt(mu0*epsilon0));
epsilon_rc = (epsilon_r)-(1j*(sigma/(omega*epsilon0)));
epsilon_complex = (epsilon0*epsilon_r)-(1j*(sigma/omega));
epsilon1 = abs(epsilon_complex);
k_1 = omega*(sqrt(mu0*epsilon1));
% Performing Integration:
k_t = sym('k_t');
row = sym('row');
k_z0 = sqrt(((k_0)^2)-((k_t)^2));
k_z1 = sqrt(((k_1)^2)-((k_t)^2));
Z_0 = ((k_z0)/(omega*epsilon0));
Z_1 = ((k_z1)/(omega*epsilon1));
Gamma_TM = (((Z_1)-(Z_0))/((Z_1)+(Z_0)));
intermediate1 = (1/(k_z0))*(1-(Gamma_TM))*(exp(-1j*(k_z0)*h))*((k_t)^3);
intermediate2 = (k_t)*row;
J = besselj(0,intermediate2);
intermediate3 = int(J*intermediate1,k_t,0,inf);
intermediate4 = ((-(Il)/(4*pi))*(1/(omega*epsilon0))*(intermediate3));
E_z = abs(intermediate4);
I need to find out the expression for "intermediate4" and E_z. Alos please let me know how I would be able to plot a graph between "intermediate4" (or E_z) and row !
P.S: Please make sure your answers are as detailed and explicit as possible !
If you feel that my question is not clear, please take a look at the attached document that contains the integral to be evaluated in highlighted form. All the other variables except k_t and "row " (greek letter) are known. I am supposed to evaluate the integral with respect to k_t from 0 to infinity and plot E (row, zero) by varying "row" !
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