A short magnetic dipole when placed at origin with its dipole moment oriented along x-axis , the magnetic field at the point (a,0) is `vecB` . What will be the magnetic field at the above point , if another identical dipole is also placed at origin with its dipole moment oriented along y-axis ?

Explanation

The initial situation is shown in figure-1 .

Clearly `vecB = ((mu_(0))/(4pi) (2M)/(a^(3)))hat i`
The final situation is shown in figure-2 . Now the field at the given point is `vecB_("Net") = vecB + vecB'`
`vecB' = - ""(-mu_(0))/(4pi) (M)/(a^(3)) hatj`
`= (mu_(0))/(4pi) (M)/(a^(3)) (=-hatj)`
`therefore vecB_("Net") = vecB + vecB'`
`= (mu_(0))/(4pi) (2M)/(a^(3)) hati + ((mu_(0))/(4pi) (M)/(a^(3))) (-hatj)`
`|vecB_("Net")| = sqrt(((mu_(0))/(4pi) (2M)/(a^(3)))^(2) + ((-mu_(0))/(4pi) (M)/(a^(3)))^(2))`
`= (mu_(0))/(4pi) (M)/(a^(3)) sqrt5`