emf.ipynb

Kernel: Python 3 (Anaconda)

import sympy as sp
sp.init_printing()

sp.var('B,L, m, F_B, E, R, t')
v = sp.Function('v')
I = sp.Function('I')
F_B = B * L *I(t) 
eq1 = sp.Eq(v(t).diff(t),F_B/m)
eq2 = sp.Eq(I(t), (E - B * L * v(t)) / R)
eq1, eq2

$$\left ( \frac{d}{d t} v{\left (t \right )} = \frac{B L}{m} I{\left (t \right )}, \quad I{\left (t \right )} = \frac{1}{R} \left(- B L v{\left (t \right )} + E\right)\right )$$

eq3  = eq1.subs([(I(t),eq2.rhs)])
eq3

$$\frac{d}{d t} v{\left (t \right )} = \frac{B L}{R m} \left(- B L v{\left (t \right )} + E\right)$$

sp.dsolve(eq3,v(t))

$$v{\left (t \right )} = \frac{1}{B L} \left(E + e^{B L \left(- \frac{B L t}{R m} + C_{1}\right)}\right)$$

sp.solve(_.rhs.subs([(t,0)]),'C1')

$$\left [ \frac{1}{B L} \log{\left (- E \right )}\right ]$$