#!/usr/bin/env python
# coding: utf-8

# # 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))))


# In[5]:


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


# In[6]:


Math(sym.latex(factor))


# 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,))))


# 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))))


# 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))))


# In[11]:


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


# In[12]:


sym.latex(factor)


# In[13]:


Math(sym.latex(factor))


# In[14]:


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


# In[15]:


sym.latex(factor)


# In[16]:


Math(sym.latex(factor))


# In[17]:


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


# In[18]:


sym.latex(factor)


# In[19]:


Math(sym.latex(factor))


# 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}