Ampere's Law Exercise

I shall show the line integral calculation steps for $Q_{12}$ .

Any point $P$ on the line joining points $P_1$ and $P_2$ can be parameterised as follows:

$\vec{r}_p = \vec{P}_1 + L \frac{\vec{P}_2 - \vec{P}_1}{\lvert \vec{P}_2 - \vec{P}_1 \rvert}$

Where the parameter $L$ varies from zero to $a_1 = \lvert \vec{P}_2 - \vec{P}_1 \rvert$ . Any point on the circular loop can be parameterised using polar coordinates:

$\vec{r}_c = \cos{\theta} \hat{i} + \sin{\theta} \hat{j}$

$d\vec{r}_c = \left(-\sin{\theta} \hat{i} + \cos{\theta} \hat{j}\right)d\theta$

Applying Biot Savart law to obtain the field at point $P$ on the line joining $P_1$ and $P_2$ gives:

$d\vec{B}_1 = \frac{\mu_o I}{4\pi} \left(\frac{d\vec{r}_c \times \left(\vec{r}_p-\vec{r}_c\right)}{\lvert \vec{r}_p-\vec{r}_c \rvert ^3}\right)$

The total magnetic field at point $P$ is, therefore:

$\vec{B}_1 = \int_{0}^{2\pi} d\vec{B}_1$

A line element on the line joining points $P_1$ and $P_2$ is as follows:

$d\vec{r}_p = dL \frac{\vec{P}_2 - \vec{P}_1}{\lvert \vec{P}_2 - \vec{P}_1 \rvert}$

The line integral is therefore:

$Q_{12} = \int_{0}^{a_1} \vec{B}_1 \cdot d\vec{r}_p$

This calculation is done using a simple script of code as follows. The other line integrals can follow the exact same procedure. Bear in mind the directions of current flow at all times.

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r1 = [0;0;0];
r2 = [1;1;1];
r3 = [2;-1;-1];
% Direction of current in line joining P1 and P2:
r12 = r2 - r1;

% Length element on line:
dL12 = 1e-3;

% Initialisation:
Q12 = 0;
for L12 = 0:dL12:norm(r12)

    % Position vector of point P on the considered line segment:
    rp = r1 + L12*(r12/norm(r12));

    % Length element of considered line in the vicinity of P
    drp = dL12*(r12/norm(r12));

    % Initialisation of magnetic field:
    B1 = 0;

    % Angle step:
    dtheta = pi/1000;

    % Obtaining magnetic field at point P:
    for theta = 0:dtheta:2*pi

        % Position vector of point on circular loop and arc length element:
        rc = [cos(theta);sin(theta);0];
        drc = [-sin(theta);cos(theta);0]*dtheta;

        % Biot-Savart Law:
        dB  = (1/(4*pi))*cross(drc,(rp-rc))/(norm(rp-rc)^3);

        % Numerical Integration:
        B1  = B1 + dB;
    end

    % Evaluation of line integral:
    % Elementary dot product:
    dQ12 = B1'*drp;
    % Numerical Integration:
    Q12   = Q12 + dQ12;
end
% Q12 = 0.4455;

% Same calculation procedure for computing Q23 and Q31:

% Q23 = 0.0752;
% Q31 = 0.4811;