Electric Potential of a Dipole & a System of Charges | Physics for JEE Main & Advanced PDF Download

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 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
  Now ,
  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:
  
  for r>>a
• 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 r_1P is the distance between q₁  and _P. 
• Similarly, the potential V₂ at P due to q₂ and due to q are given by 
  where r_2P and r_3P 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