## Potential due to an electric dipole

- We already know that 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 r_{1} and r_{2} 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

Now ,

r_{1} = QR≅RD = OR-OD = r-acosθ

r_{2} = 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

for r>>a

- From above equation we can see that potential due to electric dipole is inversly proportional to r
^{2} not 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**.

**Question 1**

What is the electric potential due to an electric dipole at an equatorial point?

**Solution**

Zero, as potential on equatorial point, due to charges of electric dipole, are equal in magnitude but opposite in nature and hence their resultant is zero.

**Question 2**

A short electric dipole has dipole moment of $4 \times 10^{-9}$ Cm. Find the following

(a) Electric Potential at a point distanct .3m from center of the dipole on the axial line

(b) Electric Potential at a point distanct 1 m from center of the dipole on the equatorial line

(c) Electric Potential at a point distanct .3m from center of the dipole on an line making at angle 30° with the dipole axis

**Solution**

Given

p=$4 \times 10^{-9}$ Cm

Potential of diple is given as

$V= \frac {1}{4 \pi \epsilon _0} \frac {p cos \theta}{r^2}$

(a) $\theta =0^0$, r=.3m

$V= \frac {1}{4 \pi \epsilon _0} \frac {p}{r^2}= 400 V$

(b) $\theta =90^0$, r=1m

$V=0$

(c) $\theta =30^0$, r=.3m

$V= \frac {1}{4 \pi \epsilon _0} \frac {p cos \theta}{r^2}$

Subsituting the values

$V=200 \sqrt 3$ V

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