Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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2d40002     1-D Superposition of dipole fields along axis     2d40002
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Question

 

Two dipoles are oriented as shown below. One dipole is made of opposite charges of magnitude , and the other dipole is made of opposite charges of magnitude . Both dipoles have a separation . Derive an equation for the net electric field at the point shown if the distance is much larger than the charge separation . (i.e. )


Figure: Two dipoles.

 



Solution

 

Sketch a picture of the electric field at the given point due to to each dipole, and sketch the net electric field.


Figure: Net electric field due to two dipoles, along the axis of the dipoles.

The electric field due to a dipole along the axis of the dipole, if the distance is much greater than the charge separation , is

The electric field a distance from the first dipole has a magnitude

Expressed as a vector,

Similarly, the electric field a distance from the second dipole has a magnitude

Expressed as a vector,

The net electric field is given by the Superposition Principle:

Note that it has a positive x-component, thus agreeing with our initial sketch of the net electric field at the given point.