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**1. 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.

**2. 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)

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

Electric field of a dipole in spherical coords

E={p/4*pi*ε0*r

^{3}}{2cosθ

__r(hat)__+sinθ

__θhat__}

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

F=m[itex]\frac{d^{2}x_{i}}{dt^{2}}[/itex]

F=qE

**3. 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 couldnt 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:

[itex]\ddot{r}[/itex]=[itex]\frac{2Acosθ}{r^{3}}[/itex]

[itex]\ddot{θ}[/itex]=[itex]\frac{Asinθ}{r^{3}}[/itex]

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!

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