# Apply a binomial approximation for a three dimensional potential field

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Consider a dipole consisting of two charged particles on the x axis, one with positive charge q+ located at x=+1/2d and one with negative q- charge located at x=-1/2d

a) Derive an exact expression for the three dimensional potential field created by this dipole at a point P whose coordinates are {x, y, and z}

b) Use the binomial approximation to show that for points who distance r from the origin is very large compare to d, the electric potential is approximately given by

(x,y,z)=(kqd/r^2)(x/r)=(kqd/r^2)(cos)

where (cos) =x/r is the angle that the position vector of point P makes with the x axis. Note that the potential of a dipole falls off as 1/r^2 compared to 1/r for a point charge. Note also that it is a lot easier to compute the electric field of a dipole using this formula that it is to calculate the field directly.

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General expression for potential of the electrical field of two opposite charges + q and - q is

,

where r+ = (r2 + (d/2)2 - 2r(d/2)cos)1/2; r-= (r2 + (d/2)2 + 2r(d/2)cos)1/2; and r - is a distance from the dipole's ...

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