Electric Field in a Dipole falls off as r^3?

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Zealot153
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Electric Field in a Dipole falls off as r^3?

Postby Zealot153 » Mon Mar 02, 2009 10:15 pm UTC

Hey guys,

Spring 2002, 8.02 Physics (E&M) from MIT OCW: http://ocw.mit.edu/OcwWeb/Physics/8-02E ... mbed02.htm

In the lecture, at around 23:00, Lewin says that the electric field around the dipole falls off as r^3, as opposed to r^2.
Context: When very far away from a +3 and -1 charge, you can view the two charges as just a +2 charge in between them somewhere, and Coloumb's Law holds (inverse distance relationship). With a +1 and -1 charge, the field falls off as an inverse cube relationship.

Lewin says the students have to prove it for homework, but it's not in the online solutions. Can anyone explain why the field falls off as an inverse cube relationship?

Thanks!

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cpt
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Re: Electric Field in a Dipole falls off as r^3?

Postby cpt » Mon Mar 02, 2009 11:09 pm UTC

http://hyperphysics.phy-astr.gsu.edu/hb ... ipole.html
The second section is about potential. Take the negative gradient to find the field.

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Charlie!
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Re: Electric Field in a Dipole falls off as r^3?

Postby Charlie! » Mon Mar 02, 2009 11:13 pm UTC

You can probably do this yourself :)

Imagine a one-dimensional setup, with a plus charge at x=1 and a minus charge at x=-1. What's the magnitude of the field some distance d away (with d bigger than 0.5)? It's easy to write as two fractions. Now try combining all of it into a single fraction by multiplying across, and I bet you'll see the 1/d3.
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Zealot153
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Re: Electric Field in a Dipole falls off as r^3?

Postby Zealot153 » Mon Mar 02, 2009 11:49 pm UTC

Hey! Thanks cpt and Charlie! I see now that the K*d^2's cancel - wow! It works out so nicely. Thanks guys!

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danpilon54
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Re: Electric Field in a Dipole falls off as r^3?

Postby danpilon54 » Tue Mar 03, 2009 1:50 am UTC

Try reading about multipole expansions.

http://en.wikipedia.org/wiki/Multipole_expansion

The electric field from a point charge drops off as r^2. A pure dipole has no net charge, so acts differently. You will find this extends to 2^n poles.
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Zealot153
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Re: Electric Field in a Dipole falls off as r^3?

Postby Zealot153 » Tue Mar 03, 2009 5:01 am UTC

Unfortunately, I only know single-variable Calc (I'm a high school student trying to teach myself E&M b/c our school doesn't offer it). That stuff looks crazy - I'll make sure to read up on it when I learn the necessary math. Thanks!

Rentsy
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Re: Electric Field in a Dipole falls off as r^3?

Postby Rentsy » Wed Mar 04, 2009 5:38 am UTC

A magnetic field also falls off at the cube of the distance.

A monopole would fall off the square.

The fact that a magnet is basically two monopoles causes them to fall off faster, especially because the farther away you get, the ratio of the distance between the two poles to the distance from one pole to the target object becomes very small.


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