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dmistry
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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)
[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*r3}{2cosθr(hat)+sinθθhat}
[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
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:
[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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