Implementation of the analytical expression for the magnetic field ... (2024)

You define beta2 as a.^2 + r.^2 + 2.*a.*r, but the printed text has the 2ar term multiplied by sin(theta) in a similar manner to that for alpha2. (I've no idea if that will solve your problem though!).

Thanks, I updated the code accordingly!

%% Analytical calculation of the magnetic field of the Helmholtz coil arrangement %%

% Approximation: The coil diameter is neglected. All windings "in one

% place"

% approximation: The magnetic table top is assumed to act as a perfect

% magnetic "mirror" is assumed.

%

format compact;

% Radius of the Coil in meters:

a = 0.2;

% Current through Coil in amperes:

I = 5.0;

% Number of Coil windings:

N = 154; % source: datasheet Helmholtz coils

r_test = sqrt(0.2.^2+0.2.^2);

[B_r1,B_theta1] = magneticField_circularCoil(I,N,a,r_test,45.0)

[B_r2,B_theta2] = magneticField_circularCoil(I,N,a,r_test,135.0)

function [B_r,B_theta] = magneticField_circularCoil(I,N,a,r,theta)

%MAGNETICFIELDCOMPONENTS Calculates the magnetic field components B_r and

%B_theta (spherical coordinates)

% B_r: B component in r direction

% B_theta: B component in theta direction

% I: current through conductor

% N: number of coil windings

% a: radius of the coil

% r: distance from the origin (spherical coordinates)

% theta: angle to z-axis (spherical coordinates) IN DEGREES

%

% Source for used analytic formula:

% https://ntrs.nasa.gov/api/citations/20140002333/downloads/20140002333.pdf

mu0 = 4.*pi.*1e-7;

alpha2 = a.^2 + r.^2 - 2.*a.*r.*sind(theta);

beta2 = a.^2 + r.^2 + 2.*a.*r.*sind(theta);

k2 = 1 - alpha2./beta2;

C = mu0 * I./pi;

[K_k2,E_k2] = ellipke(k2);

B_r = N.*(C.*a.^2.*cosd(theta))./(alpha2.*sqrt(beta2)) .* E_k2;

B_theta = N.*C./(2.*alpha2.*sqrt(beta2).*sind(theta)) .* ((r.^2+a.^2.*cosd(2.*theta)).*E_k2 - alpha2.*K_k2);

B_phi = 0;

end