A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60°, and one of the fields has a magnitude of `1.2 xx 10^(-2) T.` If the dipole comes to stable equilibrium at an angle of 15° with this field, what is the magnitude of the other field?

Explanation

Here angle between `overset(to) (B_1) and overset(to)(B_2)` is `theta =60^@`.
Suppose, `overset(to)(B_R) = overset(to) (B_1) + overset(to) (B_2)`
Now, as per the statement if `B_(1) = 0.012 T`, then angle between `overset(to)(m) ("or" B_(R) ) and overset(to)(B_1)` will be `alpha= 15^(@)`. (Here ,` overset(to)(m)` is the dipole moment of given magnetic dipole).
For stable equilibrium, we know that `overset(to) (m) || overset(to) (B_R)` .
See the figure given below. Here,
`angle SPR = angle SPQ - angle RPQ`
`therefore beta = 60^@ - 15^(@) = 45^(@)`

When magnetic dipole comes under stable equilibrium, torques exerted on it by `B_(1) and B_(2)` are equal in magnitude and opposite in direction. Hence,
`tau_(1) (xx) = tau_(2) (.)`
`mB_(1)sin alpha = mB_(2) sin beta`
`therefore B_(2) = (B_(1) sin alpha )/( sin beta) = ((0.012)sin (15^(@ ) ) )/( sin (45^(@) ) )= ((0.012 )(0.2588) )/( (0.7071) )`
`therefore B_(2) = 0.004392` T