Electromagnetism - Electric Potential

Electric Potential

The amount of work done to bring a unit positive charge from infinity to that point against the electric field of a given charge without charging

Kinetic energy: V = ∫r ∞ - E dx = Q/4πε₀r it is a scalar quantity and its unit is volt. 

1 volt = 1j/1C

The dimensions are MI L₂ T₃ A^-1

Potential difference 

?V = v₂ - v₁ = ∫r₂ r₁ - E dr

= Q / 4πε₀r|r₂r₁ = Q / 4?ε₀ [ 1/r₂ - 1/r₁] 

Equipotential surface is the surface, where the potential is equal at every point. For a point charge, a sphere will be equipotential surface with charge at the centre of the sphere. Equipotential surface for a long line charge is cylinder within charge along axial line, for a dipole, equipotential surface is a plane passing through equatorial line. 

The work done in carrying a charge form one pint to another in a equipotential surface is zero. 

The electric lines of force are always perpendicular to the equipotential surface.

Every conductor is an equipotential surface as electric filed E is perpendicular to it. 

Electric field along the equipotential surface is zero these surfaces do not intersect each other. 

E = - ?V

Electric field and surface charge density are maximum at pointed ends.

Electric field intensity due to a shell

E _inside = 0                                     x < R

E _surface = Q/4πε₀R²                  x = R

E _outside = Q/4πε₀x²                   x > R

Electric potential due to a shell

V _inside = Q / 4πε₀R = V _surface            x < R

V _surface =Q / 4πε₀R                                x = R

V _outside   = Q / 4πε₀x                              x > R

Electric filed intensity due to a dies of radius r having surface charge density σ at a point p, distant x on the axial line is

E = σ/2ε₀ [ 1 - x / x² + r²]    = σ / 2ε₀ if r ---> ∞

Or x --> 0, that is at the centre of the disc

Electric potential V = (σ/2ε₀) [x² + r²   - x] 

Dipole, moment p = q (2I) the direction is along negative to positive side.

Electric filed intensity due to a dipole
    
Along axial line

E_axial = 2px / 2ε₀ (x² - I² )²

For a short dipole x >>I E axial = 2p / 4πε₀x³


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