Motion of an electric charge in the field of an electric dipole

[dmistry]

Homework Statement

An electric dipole is situated at the origin and points along z. An electric charge is
released from rest at a point in the (x,y) plane. Show that it swings back and forth in a
semicircular arc about the origin.

Homework Equations

Electric field of a dipole (coord free- I tried using this one but didnt get anywhere, I think using form in spherical coords is more useful)

\vec{E}(r)=\frac{3(\vec{p}.\hat{r})\hat{r}-\vec{p}}{4\pi\epsilon_{o}r^{3}}

Electric field of a dipole in spherical coords

E={p/4*pi*ε0*r³}{2cosθr(hat)+sinθθhat}

\vec{E}=\frac{p}{4\pi\epsilon_{o}r^{3}}(2cosθ\hat{r}+sin(θ)\hat{θ})

F=m\frac{d^{2}x_{i}}{dt^{2}}

F=qE

The Attempt at a Solution

With each form of the dipole E field I used F=qE and equated it to the ma expression and tried to solve the differential equation for each direction (x,y,z) or (r,theta,phi). It didnt work for the coord free form, well I couldn't get it to work dues to the r^3 term. More promising was using the spherical coord form where I ended up with two coupled differential equations:

\ddot{r}=\frac{2Acosθ}{r^{3}}

\ddot{θ}=\frac{Asinθ}{r^{3}}

but I have no idea how to solve them. I'm pretty sure I'm wanting a complex exponential solution of sorts as this would obviously fulfil the oscillating motion, but I have no idea if those differential equations are correct or how to solve them.

Any help would be greatly appreciated!

 

[SImple66]

To me you have pretty much solved the problem the goal was to show that the particle oscillates in a semicircle like orbit which your θ double dot shows from it containing a sin(θ) in it. The θ double dot is the acceleration up and down and a sin wave goes from positive to negative to positive and so on and so forth. This means the particle is flying one way then another then back to where it came from. It just goes back and forth. So what you have here works

 

[TSny]

\ddot{r}=\frac{2Acosθ}{r^{3}}

\ddot{θ}=\frac{Asinθ}{r^{3}}

SImple66, Welcome to PF!

Note that the expressions for the acceleration components on the left hand sides of dmistry's equations quoted above are not correct.

See https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=dy&chap_sec=01.6&page=theory for a review of velocity and acceleration in polar coordinates.

Once you have the left hand sides corrected, you just need to show that the differential equations with the stated initial conditions can be solved with a motion along a circular arc. That is, assume the motion is along a circular arc and show that this type of motion satisfies the differential equations. (You do not need to find an explicit solution for how θ varies with time.) A uniqueness theorem guarantees that there is only one solution that satisfies the initial conditions.

This is a very interesting result! Thanks for bringing attention to this old thread.

 

[Safakphysics]

This question solved by use analogy between simple pendulum that pivotted from one point.

 

[vela]

This thread was started four years ago, so I doubt the OP is still around. I've moved it to the old-homework forum.