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There appears to be two methods we can use to calculate the force (effectively the same in fact, but one is a bit of a shortcut):

1. Find the potential energy of the dipole using

U=-

**p**.

**E**

=-p

_{x}E

_{x}-p

_{y}E

_{y}-p

_{z}E

_{z}.

Then simply use

**F**=-

**∇**U to get the force. This comes out as

**F**=(p

_{x}∂E

_{x}/∂x+p

_{y}∂E

_{y}/∂x+p

_{z}∂E

_{z}/∂x)

**i**+(p

_{x}∂E

_{x}/∂y+p

_{y}∂E

_{y}/∂y+p

_{z}∂E

_{z}/∂y)

**j**+(p

_{x}∂E

_{x}/∂z+p

_{y}∂E

_{y}/∂z+p

_{z}∂E

_{z}/∂z)

**k**

Now this method has given me correct answers when I have started with a specific vector function

**E**, found its corresponding U and then got

**F**

2. The 'shortcut'. We can use a vector calculus identity so that

**F**=-

**∇**U=-

**∇**(-

**p**.

**E**)=

**∇**(

**p**.

**E**)=(

**p**.

**∇**)

**E**. All this means is we can find

**F**without the intermediate stage of needing U. All we need is

**E**. However this in component form gives

**F**=(p

_{x}∂E

_{x}/∂x+p

_{y}∂E

_{x}/∂y+p

_{z}∂E

_{x}/∂z)

**i**+(p

_{x}∂E

_{y}/∂x+p

_{y}∂E

_{y}/∂y+p

_{z}∂E

_{y}/∂z)

**j**+(p

_{x}∂E

_{z}/∂x+p

_{y}∂E

_{z}/∂y+p

_{z}∂E

_{z}/∂z)

**k**

So the same vector

**F**comes out in two different ways. Now, I can't find any reason that makes them equal, but I recall using the first correctly for a specific field, and the second method I'm assuming would have given me the correct answer too, so why aren't they agreeing with one another for a general dipole in a general field in component form? Thanks for any help!