If the current through a conducting loop in the shape of an equilaterla triangle of side a be I, what will be the magnitude of the magnetic field of intersection of the three medians ?

Explanation

Lengths of each median of the triangular loop
`= sqrt(a^(2)-((a)/(2))^(2))=(sqrt(3))/(2)a`
Hence the distance of each side from the point of intesection O of the median [Fig. 1.89]
`r=OA=(1)/(3)*(sqrt(3))/(2)a=(a)/(2sqrt(3))`
Angle subtended by the two extremities of each side at O, `theta_(1) = theta_(2) =60^(@)`.
So due current I through each side, magnetic field at the point O,
`B=(mu_(0))/(4pi) *(I)/(r)(sin theta_(1) + theta_(2))`
`=(mu_(0))/(4pi)*(I)/((a)/(sqrt(3)))(sin 60^(@)+sin 60^(@))`
`=(mu_(0))/(4pi)*(2sqrt(3)I)/(a)*2 xx (sqrt(3))/(2)=(mu_(0))/(4pi)*(6I)/(a)`

For current flowing through each side, the direction the direction of the magnetic field at the point O is indetical and hence, the resultant magnetic field at the point `O=3B=(mu_(0))/(4pi)*(18I)/(a)`