A bar magnet placed in a uniform magnetic field making an angle `theta` with the field experiences a torque. If the angle made by the magnet with the field is doubled, the torque experienced by the magnet increases by`41.4%`. The initial angle made by the magnet with the magnetic field is

Options

A. `60^(@)`
B. `30^(@)`
C. `90^(@)`
D. `45^(@)`

Answer

D

Explanation

As we know, torque in a magnetic field,
`tau = MB sin theta`
if ` theta = theta_(1),`
then `tau_(1) = MB sin theta _(1)" "...(i)`
Similarly, if `theta = theta_(2)` then
`tau_(2) = MBsin theta_(2) = MB sin 2 theta_(1) " " (because" Given, " theta_(2) = 2 theta _ (1))`
Given `therefore tau_(2) = tau_(1) + tau_(1) xx (41.1)/100`
`= 1.414 tau_(1) = sqrt(2)tau_(1)`
`rArr sqrt(2) tau_(1) = MB sin 2 theta_(1)" " ... (ii)`
From Eqs. (i) and (ii) , we get
`1 /sqrt(2) = sin theta _(1)/(sin 2 theta_(1))`
`rArr sin 2 theta_(1) = sqrt(2 ) sin theta_(1)" "...(iii)`
As we know that `sin 2 theta = 2 sin theta cos theta`
Hence, `2sin theta_(1) cos theta_(1) = sqrt(2) sin theta_(1)`
`2 cos theta = sqrt(2) rArr cos theta = 1/sqrt(2) rArr theta = 45^(@)`
Hence, the correct option is (d).