Electric Potential

Electric Potential Energy

ΔU=-W

Where ΔU = Change in Potential energy
W= Workdone by the electric lines of forces

For a system of two particles

U(r)=q₁q₂/4πεr

where r is the seperation between the charges

We assume U to be zero at infinity

Similarliy for a system of n charges
U=Sum of potential energy of all the distinct pairs in the system

For example for three charges
U=(1/4πε)(q₁q₂/r₁₂+q₂q₃/r₂₃+q₁q₃/r₁₃)

Another way to represent
U=1/2ΣqV

where V is the potential at charge q due to all the remaining charges

Electric Potential:
Just liken Electric field intensity is used to define the electric field,we can also use Electric Potential to define the field

Potential at any point P is equal to th workdone per unit test charge by the external agent in moving the test charge from the refrence point(without Change in KE)

V_p=W_ext/q

So for a point charge

V_p=Q/4πεr

where r is the distance of the point from charge

Some points about Electric potential1. It is scalar quantity
2.Potential at point due to system of charges will be obtained by the summation of potential of each charge at that point

V=V₁+V₂+V₃+V₄

3.Electric forces are conservative force so workdone by the electric force between two point is independent of the path taken

4. V₂-V₁=-∫ E.dr

5 In cartesion coordinates system
E=E_xi+E_yj+E_zk
dr=dxi+dyj+dzk

Now
dV=-E.dr

So dv=-(E_xdx+E_ydy+E_zdz)

So E_x=∂V/∂x

Similary
E_y=∂V/∂y and E_z=∂V/∂z

Also
E=-[(∂V/∂x)i+(∂V/∂y)j+(∂V/∂z)k]

4
Surface where electric potential is same everywhere is call equipotential surface
Electric field components parallel to equipotential surface is always zero



Electric dipole:
A combination of two charge +q and -q seperated by the distance d

p=qd
Where d is the vector joiing negative to positive charge

Electric potential due to dipole
V=(1/4πε)(pcosθ/r²)

where r is the distance from the center and θ is angle made by the line from the axis of dipole

Electric field
E_θ=(1/4πε)(psinθ/r³)

E_r=(1/4πε)(2pcosθ/r³)

Total E=√E_θ²+E_r²
=(p/4πεr³)(√(3cos²θ+1))

Torque on dipole=pXE

Potential Energy
U=-p.E