Conducting Objects Having Sharp Points

Consider two conducting spheres of radii R₁ and R₂, placed in contact of each other. When a charge is given to this system, it rises to a common potential V. If q₁ and q₂ be the charges on the two spheres respectively, then we have

    V = 1/4πε₀ q₁/R₁ = 1/1/4πε₀ q₂/R₂
. :     q₁/q₂ = R₁/R₂

If σ₁ and σ₂ be the charge densities of the two spheres, we have
        σ₁ = q₁/4πR₁² and σ₂ = q₂/4πR₂²
    . :    σ₁/ σ₂ = R₁/R₂                                    … (1)

Substituting the value of q1/q2 from Eq. (1), we get
        σ₁/ σ₂ = R₂/R₁
or        σ∞ 1/R
That is, smaller the radius of the sphere, higher is the charge density.
Thus if a conducting object has sharp points on its surface, the charge densities and hence the electric intensities near these points will be very high. These points will strongly attract the oppositely-charged ions of the surrounding air. These ions will therefore be accelerated and produce more ions by colliding with the air molecules. The air will thus become quite conducting giving rise to corona discharge. The air surrounding sharp conducting points may even glow by the light emitted from the air molecules during the collisions.