1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Equation of motion: Help with DiffEq (2nd order non linear)

  1. Aug 20, 2015 #1
    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!
     
  2. jcsd
  3. Aug 20, 2015 #2

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You can do it in three stages:

    1) First, multiply by ##r'## (integrating factor) to get:

    ##\frac{d}{dt}(r'^2) = \frac{d}{dt}(\frac{-2C}{r})##

    2) (The key trick): Let ##r = Dcos^2\theta##

    This leads to:

    ##(cos^2\theta)\theta ' = \sqrt{\frac{-C}{2D^3}}##

    3) Integrate to get:

    ##2\theta + sin(2\theta) = \sqrt{\frac{-8C}{D^3}}t##
     
  4. Aug 20, 2015 #3

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    2017 Award

    So no simple solution. I typed x'' = -1/x^2 here and it came back with this picture:

    http://www4c.wolframalpha.com/Calculate/MSP/MSP12181g1a4ceci2e2i16500005a8e262bb5i0g8e3?MSPStoreType=image/gif&s=10&w=388.&h=101. [Broken]
    Got the same picture doing a simple numeric integration with excel:

    upload_2015-8-20_22-29-40.png

    (x on the left axis, v and a on the right axis)


    Basically the mobile charge just "falls" towards the fixed charge in the same way a small mass falls towards a planet: initially slowly (the change in attractive force is small) and then accelerating faster and faster.
     
    Last edited by a moderator: May 7, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook