- #1
Adoniram
- 94
- 6
I am trying to solve the differential equation that will give me the equation of motion of a point charge under the influence of another point charge's electric field.
Say point charge A is free to move, and it currently a distance D away from point charge B. Point charge B is fixed in space.
Say charge A has q = +q, and charge B has q = -q. The two charges will attract.
Ignoring all other influences (gravity, etc), charge A should experience a force F = qE, where E is the field due to charge B, or:
F = -(k q^2)/r^2
where k = 1/4πε (imagine that ε is the permittivity of free space; I'm using the available symbols)
Solving for equations of motion, I use:
ma = -(k q^2)/r^2
or
a = -(k q^2)/(m r^2)
Putting it another way, r → r[t], a → r''[t]
Then I get:
r[t]2 r''[t] = -(k q^2)/m
Or
r[t]2 r''[t] = C
How do I solve that differential equation? It is a 2nd order non-linear diff eq... The Mathematica answer I get is very complicated, but I'm hoping someone can help me out with this one.
Also, I can use the following initial conditions:
r[0]=D
r'[0]=0
Thanks!
Say point charge A is free to move, and it currently a distance D away from point charge B. Point charge B is fixed in space.
Say charge A has q = +q, and charge B has q = -q. The two charges will attract.
Ignoring all other influences (gravity, etc), charge A should experience a force F = qE, where E is the field due to charge B, or:
F = -(k q^2)/r^2
where k = 1/4πε (imagine that ε is the permittivity of free space; I'm using the available symbols)
Solving for equations of motion, I use:
ma = -(k q^2)/r^2
or
a = -(k q^2)/(m r^2)
Putting it another way, r → r[t], a → r''[t]
Then I get:
r[t]2 r''[t] = -(k q^2)/m
Or
r[t]2 r''[t] = C
How do I solve that differential equation? It is a 2nd order non-linear diff eq... The Mathematica answer I get is very complicated, but I'm hoping someone can help me out with this one.
Also, I can use the following initial conditions:
r[0]=D
r'[0]=0
Thanks!