In vacuum, three small spheres are placed on the circumference of a circle of radius r in such a way that an equilateral triangle is formed. If q be the charge on each sphere, determine the intensity of electric field and potential at the centre of the circle.

Explanation

Let O be the centre of the circle. Three spheres are placed at the points A, B and C on the circumference of the circle, ABC is an equilateral triangle.
Intensity at O due to the charge at A,
`E_(A) = q/(4pi epsilon_(0)r^(2)) , along vec(AO)`
Intensity at O due to the charge at B,
`E_(B) = q/(4 pi epsilon_(0)r^(2)) , along vec(BO)`
Intensity at O due to the charge at C,
`E_(C ) = q/(4 pi epsilon_(0)r^(2)) , along vec(CO)`
Resolving the above intensities along AO and perpendicular to AO we have,
intensity along `AO = E_(A) - E_(B) cos 60^(@) - E_(C) cos 60^(@)`
`=1/(4 pi epsilon_0) [q/(r^2) - q/(r^2) xx 1/2 - q/(r^2) xx 1/2] = 0`
Intensity perpendicular to AO
`E_(A) cos 90^(@) + E_(B) sin 60^(@) -E_(C) sin 60^(@)`
`1/(4 pi epsilon_(0))[0+ q/(r^2) sin 60^(@) - q/(r^2) sin 60^(@)] = 0`
`:.` Resultant intensity at O = 0
and electric potential at `O = 1/(4 pi epsilon_(0))[q/r + q/r +q/r] = (3q)/(4 pi epsilon_(0)r) units`.