### Electric Dipole - Potential At a Point Due To Electric Dipole.

- An electric dipole consists of two point charges of equal magnitude but of opposite sign and separated by a small distance.

- Consider an electric dipole centered at origin and placed in z – axis as shown in the figure:

- The potential (V) at point P is given as:

- If the distance between the charges (d) is very small as compared to the distance of the point P from the origin i.e.

If r >> d,

r2 – r1 d cosθ ;          r1 ≅ r2 = r ;           r1r2 ≅ r2

Substituting the values in the above equation, the potential at point P becomes:

- Electric field intensity (E) is the negative gradient of Electric Potential (V).
Hence,

- The expressions for electric potential (V) and field intensity (E) above are only valid for a dipole centered at the origin and aligned with the z-axis.

- To determine the fields produced by any arbitrary location and alignment, we first need to define a new quantity p, called the Dipole Moment.

p = Q d

- Since the distance d is a vector quantity, the dipole moment p is also a vector quantity.

- Dipole moment p is a measure of the strength of the dipole and indicates its direction.

- Vector d is a directed distance that extends from negative charge (- Q) to positive charge (+ Q). This directed distance vector d thus describes the distance between the dipole charges, as well as the orientation of the charges.

Therefore

d = | d | ad

Where | d | is the distance between the charges and ad defines the orientation or direction of the dipole.

- Say a dipole is aligned along z – axis, then directed distance d is given as:

d = | d | az

From the above diagram it’s clear that:

az . ar = cosθ

Hence the expression can be written as:

Q d cosθ  =  Q | d | az . ar  =  Q d. ar  =  p . ar

- Hence electric potential (V) due to a electric dipole centered at origin and aligned with the z axis is rewritten as:

- The above expression no doubt is applicable for all and any dipole moments p, but is valid for dipoles centered at origin.

- Electric potential (V) at point P with a position vector r due to a dipole centered at a point with position vector r1 is given as:

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