Factoring vector equations

Magnetic due to 2-wires

General equation

\begin{align} \vec{\mathbf{B}} = {\mu}I_1 \left( - \frac{y_1}{s_1^2} \hat{x} + \frac{x_1}{s_1^2} \hat{y} \right) +{\mu}I_2 \left( - \frac{y_2}{s_2^2} \hat{x} + \frac{x_2}{s_2^2} \hat{y} \right) \end{align}

Same $y$ and $I$



In [1]:

    
import sympy as sym
from IPython.display import Math



In [2]:

    
syms = sym.symbols("\mu I1 I2 I xhat x1 x2 x yhat y s1 s2 d")
(m, I1, I2, I, xhat, x1, x2, x, yhat, y, s1, s2, d) = syms



In [3]:

    
B = m * I * (-y/s1**2*xhat + (x-d)/s1**2*yhat) + m * I * (-y/s2**2*xhat + (x+d)/s2**2*yhat)

Quick sanity check:



In [4]:

    
Math(sym.latex(sym.collect(B, (xhat, yhat))))









    Out[4]:





$$I \mu \left(- \frac{\hat{x} y}{s_{1}^{2}} + \frac{\hat{y}}{s_{1}^{2}} \left(- d + x\right)\right) + I \mu \left(- \frac{\hat{x} y}{s_{2}^{2}} + \frac{\hat{y}}{s_{2}^{2}} \left(d + x\right)\right)$$



In [5]:

    
factor = sym.factor(B, (xhat, yhat, I, m))
factor









    Out[5]:





-I*\mu*(xhat*(s1**2*y + s2**2*y) + yhat*(-d*s1**2 + d*s2**2 - s1**2*x - s2**2*x))/(s1**2*s2**2)



In [6]:

    
Math(sym.latex(factor))









    Out[6]:





$$- \frac{I \mu}{s_{1}^{2} s_{2}^{2}} \left(\hat{x} \left(s_{1}^{2} y + s_{2}^{2} y\right) + \hat{y} \left(- d s_{1}^{2} + d s_{2}^{2} - s_{1}^{2} x - s_{2}^{2} x\right)\right)$$



In [7]:

    
yhat2 = yhat*(-d*s1**2 + d*s2**2 - s1**2*x - s2**2*x)/(s1**2*s2**2)
Math(sym.latex(sym.factor(yhat2, (yhat,))))









    Out[7]:





$$\frac{\hat{y}}{s_{1}^{2} s_{2}^{2}} \left(- d s_{1}^{2} + d s_{2}^{2} - s_{1}^{2} x - s_{2}^{2} x\right)$$



In [8]:

    
yhat3 = (-d*s1**2 + d*s2**2 - s1**2*x - s2**2*x)/(s1**2*s2**2)
Math(sym.latex(sym.factor(yhat3, (s1,s2))))









    Out[8]:





$$- \frac{1}{s_{1}^{2} s_{2}^{2}} \left(s_{1}^{2} \left(d + x\right) + s_{2}^{2} \left(- d + x\right)\right)$$

From these, we get

\begin{align} \frac{s_1^2 y + s_2^2 y}{s_1^2 s_2^2} \hat{x} \end{align}\begin{align} - \frac{\Big(s_{1}^{2} \left(x + d\right) + s_{2}^{2} \left(x - d\right)\Big)}{s_1^2 s_2^2} \hat{y} \end{align}

and thus our equation for the magnetic field due to two wires with the same currents:

\begin{align} \vec{\mathbf{B}} = {\mu}I \Bigg( - \bigg( \frac{y}{s_1^2} + \frac{y}{s_2^2} \bigg) \hat{x} + \bigg( \frac{x - d}{s_1^2} + \frac{x + d}{s_2^2} \bigg) \hat{y} \Bigg) \end{align}

Different $I$



In [9]:

    
B = m * I1 * (-y/s1**2*xhat + x1/s1**2*yhat) + m * I2 * (-y/s2**2*xhat + x2/s2**2*yhat)

Sanity check:



In [10]:

    
Math(sym.latex(sym.collect(B, (xhat, yhat))))









    Out[10]:





$$I_{1} \mu \left(\frac{x_{1} \hat{y}}{s_{1}^{2}} - \frac{\hat{x} y}{s_{1}^{2}}\right) + I_{2} \mu \left(\frac{x_{2} \hat{y}}{s_{2}^{2}} - \frac{\hat{x} y}{s_{2}^{2}}\right)$$



In [11]:

    
factor = sym.factor(B, (xhat, yhat))
factor









    Out[11]:





-\mu*(xhat*(I1*s2**2*y + I2*s1**2*y) + yhat*(-I1*s2**2*x1 - I2*s1**2*x2))/(s1**2*s2**2)



In [12]:

    
sym.latex(factor)









    Out[12]:





'- \\frac{\\mu}{s_{1}^{2} s_{2}^{2}} \\left(\\hat{x} \\left(I_{1} s_{2}^{2} y + I_{2} s_{1}^{2} y\\right) + \\hat{y} \\left(- I_{1} s_{2}^{2} x_{1} - I_{2} s_{1}^{2} x_{2}\\right)\\right)'



In [13]:

    
Math(sym.latex(factor))









    Out[13]:





$$- \frac{\mu}{s_{1}^{2} s_{2}^{2}} \left(\hat{x} \left(I_{1} s_{2}^{2} y + I_{2} s_{1}^{2} y\right) + \hat{y} \left(- I_{1} s_{2}^{2} x_{1} - I_{2} s_{1}^{2} x_{2}\right)\right)$$



In [14]:

    
xhat2 = -(I1*s2**2*y + I2*s1**2*y)/(s1**2*s2**2)
factor = sym.factor(xhat2, (s1, s2))
factor









    Out[14]:





-y*(I1*s2**2 + I2*s1**2)/(s1**2*s2**2)



In [15]:

    
sym.latex(factor)









    Out[15]:





'- \\frac{y}{s_{1}^{2} s_{2}^{2}} \\left(I_{1} s_{2}^{2} + I_{2} s_{1}^{2}\\right)'



In [16]:

    
Math(sym.latex(factor))









    Out[16]:





$$- \frac{y}{s_{1}^{2} s_{2}^{2}} \left(I_{1} s_{2}^{2} + I_{2} s_{1}^{2}\right)$$



In [17]:

    
yhat2 = (I1*s2**2*x1 + I2*s1**2*x2)/(s1**2*s2**2)
factor = sym.factor(yhat2, (s1, s2))
factor









    Out[17]:





(I1*s2**2*x1 + I2*s1**2*x2)/(s1**2*s2**2)



In [18]:

    
sym.latex(factor)









    Out[18]:





'\\frac{1}{s_{1}^{2} s_{2}^{2}} \\left(I_{1} s_{2}^{2} x_{1} + I_{2} s_{1}^{2} x_{2}\\right)'



In [19]:

    
Math(sym.latex(factor))









    Out[19]:





$$\frac{1}{s_{1}^{2} s_{2}^{2}} \left(I_{1} s_{2}^{2} x_{1} + I_{2} s_{1}^{2} x_{2}\right)$$

From this, we get these:

\begin{align} - \frac{I_1 s_2^2 + I_2 s_1^2}{s_1^2 s_2^2} y \hat{x} \end{align}\begin{align} \frac{I_1 s_2^2 x_1 + I_2 s_1^2 x_2}{s_1^2 s_2^2} \hat{y} \end{align}

and thus our equation for the magnetic field due to two wires with different currents:

\begin{align} \vec{\mathbf{B}} = {\mu} \Bigg( - \bigg( \frac{I_1 s_2^2 + I_2 s_1^2}{s_1^2 s_2^2} y \bigg) \hat{x} + \bigg( \frac{I_1 s_2^2 x_1 + I_2 s_1^2 x_2}{s_1^2 s_2^2} \bigg) \hat{y} \Bigg) \end{align}\begin{align} = \frac{\mu}{s_1^2 s_2^2} \Big( - y \big( I_1 s_2^2 + I_2 s_1^2 \big) \hat{x} + \big( I_1 s_2^2 (x - d) + I_2 s_1^2 (x + d) \big) \hat{y} \Big) \end{align}