(adsbygoogle = window.adsbygoogle || []).push({}); Problem Diagram (Ignore the tildes, they're just placeholders):

Below: An electric dipole

~~~~~~~y-axis~~~~~~~~~~~~~~~~~~

~~~~~~~|~~~~~~~~~~~~~~~~~~~~~

~~~~~~~|~~~~~~~~~~~~~~~~~~~~~

~~~~~~~|~~~~~~~~~~~~~~~~~~~~~

~~<---a---> <---a--->~~~~~~~~~~~~~~~

+Q --------- X --------- -Q~~~-------------- x-axis

Problem Statement: Find the voltage at X.

My answer is that V_x = 0, since the potentials from each side of the dipole sum to zero. And I'm pretty sure this is right.But my question is... how does this fit in with the following definition of voltage:

"The voltage at an arbitrary point P is the amount of work per unit charge it takes to move a test charge from infinity to P" (Physics for Scientists and Engineers)

I see that if the test charge is approaching X from south of X, the work will be zero, since there is only a force in the east direction as the +Q and -Q cancel e/o out in the y-direction. Same thing if the test charge is approaching x from north of X.

But what if the test charge is approaching X from west of X? east of X? How is the work zero? Won't the work be infinite from the west, assuming a positive test charge, because of the asymptotic behavior of the electric field along that line? Won't it be negatively infinite from the east, assuming a positive test charge?

So, can someone explain how the work is zero coming from infinity west or east of X?

I did think to myself that the work should be path-independent... but it's not... so how would you explain this?

Thanks in advance.

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# Voltage in the Middle of an Electric Dipole

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