Class 12 : Electric Potential of a Dipole and a System of Charges Class 12 Notes | EduRev
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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.
- 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 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.
- 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