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**Potential due to an Electric Dipole**

- The electric dipole is an arrangement that 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 the 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.

A Dipole - Since electric potential obeys the 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 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 q
_{1}, q_{2},…, qn with position vectors r_{1}, r_{2},…, r_{n}relative to some origin. The potential V_{1}at P due to the charge q_{1}is:

where r_{1P}is the distance between q_{1 }and_{P}. - Similarly, the potential V
_{2}at P due to q_{2}and due to q are given by

where r_{2P}and r_{3P}are the distances of P from charges q_{2 }and q_{3}, respectively; and so on for the potential due to other charges. - By the superposition principle, the potential V at P due to the total charge configuration is the algebraic sum of the potentials due to the individual charges
- The electric field outside the shell is as if the
**entire charge is concentrated at the centre**. Thus, the potential outside the shell is given by:

where q is the total charge on the shell and R its radius. - The electric field inside the shell is zero. This implies that potential is constant inside the shell (as
**no work is done in moving a charge inside the shell**), and, therefore, equals its value at the surface, which is

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