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

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• dmistry
In summary: The solution that dmistry proposed was to assume the motion was along a circular arc and show that this type of motion satisfies the differential equations. That would indeed work.In summary, the conversation discusses finding the solution for a particle released from rest in the (x,y) plane that swings back and forth in a semicircular arc about the origin due to the electric field of a dipole. The equations for the electric field of a dipole in spherical coordinates are used along with the equations for force and acceleration to set up a differential equation. However, the solution for the differential equation is not found and it is suggested to assume the motion is along a circular arc and show that this type of motion satisfies the differential equations. This solution is found to
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!

Last edited:
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

dmistry said:
$\ddot{r}$=$\frac{2Acosθ}{r^{3}}$

$\ddot{θ}$=$\frac{Asinθ}{r^{3}}$

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.

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

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

What is an electric dipole?

An electric dipole is a pair of equal and opposite charges, separated by a small distance. It is a fundamental concept in electromagnetism and is often used to explain the behavior of electric fields and charges.

How does an electric dipole affect the motion of an electric charge?

An electric dipole creates an electric field around it, which exerts a force on any nearby electric charges. The motion of an electric charge in this field depends on its direction and magnitude.

What factors affect the motion of an electric charge in the field of an electric dipole?

The motion of an electric charge in the field of an electric dipole is affected by the distance between the charge and the dipole, the strength of the electric field, and the orientation of the dipole relative to the charge.

How does the direction of the electric dipole affect the motion of an electric charge?

If the electric dipole is aligned with the direction of motion of the charge, it will experience a greater force and will accelerate more. If the dipole is perpendicular to the motion, the force will be weaker and the charge will curve around the dipole.

Can an electric charge ever be in equilibrium in the field of an electric dipole?

Yes, if the electric charge is placed along the line connecting the two charges of the dipole, it will experience equal and opposite forces that cancel out, resulting in equilibrium. This point is known as the "electric field null point".

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