Potential due to an Electric Dipole
The electric dipole is an arrangement which consists of two equal and opposite charges +q and -q separated by a small distance 2a.
Electric dipole moment is represented by a vector p of magnitude 2qa and this vector points in direction from -q to +q.
To find electric potential due to a dipole consider charge -q is placed at point P and charge +q is placed at point Q as shown below in the figure.
Since electric potential obeys superposition principle so potential due to electric dipole as a whole would be sum of potential due to both the charges +q and -q. Thus
where r1 and r2 respectively are distance of charge +q and -q from point R.
Now draw line PC perpandicular to RO and line QD perpandicular to RO as shown in figure. From triangle POC
cosθ=OC/OP = OC/a
therefore OC=acosθ similarly OD=acosθ
r1 = QR≅RD = OR-OD = r-acosθ
r2 = PR≅RC = OR+OC = r+acosθ
since magnitude of dipole is
|p| = 2qa
If we consider the case where r>>a then
again since pcosθ= p·rˆ where, rˆ is the unit vector along the vector OR then electric potential of dipole is
From above equation we can see that potential due to electric dipole is inversly proportional to r2 not ad 1/r which is the case for potential due to single charge.
Potential due to electric dipole does not only depends on r but also depends on angle between position vector r and dipole moment p.
Potential Due To A System Of Charges
Consider a system of charges q1, q2,…, qn with position vectors r1, r2,…, r n relative to some origin. The potential V1 at P due to the charge q1 is
where r1P is the distance between q1 and P. Similarly, the potential V2 at P due to q2 and due to q are given by
where r2P and r3P are the distances of P from charges q2 and q3, respectively; and so on for the potential due to other charges.