In [18]:

    

import sympy as sp
from sympy.physics import vector as vec
sp.init_printing()
vec.init_vprinting()


    

In [19]:

    

R = vec.ReferenceFrame('R')
R


    

    
Out[19]:





R

In [3]:

    

R??


    

In [20]:

    

x, y, z = R[0], R[1], R[2]
# x, y, z, t = sp.symbols('x y z t', real=True)
t = sp.symbols('t', real=True)
c, eps, mu = sp.symbols('c varepsilon_0 mu_0')
c, eps, mu


    

    
Out[20]:





$$\left ( c, \quad \varepsilon_{0}, \quad \mu_{0}\right )$$

In [21]:

    

Ex= sp.Function('Ex')(x, t)
Ey= sp.Function('Ey')(x, t)
Ez= sp.Function('Ez')(x, t)
Bx= 0
By= sp.Function('By')(x, t)
Bz= sp.Function('Bz')(x, t)


    

In [22]:

    

E = Ex*R.x + Ey*R.y + Ez*R.z
B = Bx*R.x + By*R.y + Bz*R.z
vec.curl(E, R)


    

    
Out[22]:





$$-  \frac{\partial}{\partial R_{x}} \operatorname{Ez}{\left (R_{x},t \right )}\mathbf{\hat{r}_y} + \frac{\partial}{\partial R_{x}} \operatorname{Ey}{\left (R_{x},t \right )}\mathbf{\hat{r}_z}$$

In [31]:

    

jx= sp.Function('jx')(x, t)
jy= sp.Function('jy')(x, t)
jz= sp.Function('jz')(x, t)
j = jx*R.x + jy*R.y +jz*R.z


    

In [32]:

    

# FARADAY = sp.Eq(vec.curl(E, R), -B.diff(t,R))
FARADAY = B.diff(t, R)- vec.curl(E, R)
FARADAY


    

    
Out[32]:





$$(\frac{\partial}{\partial t} \operatorname{By}{\left (R_{x},t \right )} + \frac{\partial}{\partial R_{x}} \operatorname{Ez}{\left (R_{x},t \right )})\mathbf{\hat{r}_y} + (\frac{\partial}{\partial t} \operatorname{Bz}{\left (R_{x},t \right )} - \frac{\partial}{\partial R_{x}} \operatorname{Ey}{\left (R_{x},t \right )})\mathbf{\hat{r}_z}$$

In [36]:

    

AMPERE = vec.curl(B, R) - mu * j - eps * mu * E.diff(t, R)
AMPERE


    

    
Out[36]:





$$(- \mu_{0} \varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ex}{\left (R_{x},t \right )} - \mu_{0} \operatorname{jx}\left(R_{x},t\right))\mathbf{\hat{r}_x} + (- \mu_{0} \varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ey}{\left (R_{x},t \right )} - \mu_{0} \operatorname{jy}\left(R_{x},t\right) - \frac{\partial}{\partial R_{x}} \operatorname{Bz}{\left (R_{x},t \right )})\mathbf{\hat{r}_y} + (- \mu_{0} \varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ez}{\left (R_{x},t \right )} - \mu_{0} \operatorname{jz}\left(R_{x},t\right) + \frac{\partial}{\partial R_{x}} \operatorname{By}{\left (R_{x},t \right )})\mathbf{\hat{r}_z}$$

In [38]:

    

(FARADAY + c * AMPERE).simplify()


    

    
Out[38]:





$$-  c \mu_{0} \left(\varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ex}{\left (R_{x},t \right )} + \operatorname{jx}\left(R_{x},t\right)\right)\mathbf{\hat{r}_x} + (- c \left(\mu_{0} \varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ey}{\left (R_{x},t \right )} + \mu_{0} \operatorname{jy}\left(R_{x},t\right) + \frac{\partial}{\partial R_{x}} \operatorname{Bz}{\left (R_{x},t \right )}\right) + \frac{\partial}{\partial t} \operatorname{By}{\left (R_{x},t \right )} + \frac{\partial}{\partial R_{x}} \operatorname{Ez}{\left (R_{x},t \right )})\mathbf{\hat{r}_y} + (- c \left(\mu_{0} \varepsilon_{0} \frac{\partial}{\partial t} \operatorname{Ez}{\left (R_{x},t \right )} + \mu_{0} \operatorname{jz}\left(R_{x},t\right) - \frac{\partial}{\partial R_{x}} \operatorname{By}{\left (R_{x},t \right )}\right) + \frac{\partial}{\partial t} \operatorname{Bz}{\left (R_{x},t \right )} - \frac{\partial}{\partial R_{x}} \operatorname{Ey}{\left (R_{x},t \right )})\mathbf{\hat{r}_z}$$