A wire loop carrying `I` is placed in the ` x-y`plane as shown in fig.
(a) If a particle with charge ` +Q` and mass `m` is placed at the centre `P` and given a velocity ` vec(v)` along `NP`(see figure), find its instantaneous acceleration.
( b) If an external uniform magnetic induction field ` vec(B) = Bhat(i)` is applied , find the force and the torque acting on the loop due to this field.

Answer

A, B, C, D

Explanation

` vec(B)_(1) = (mu_(0))/(4 pi) ( 2I sqrt(3))/(a) (-hat (k)) , vec(B)_(2) = ( mu_(0))/( 4 pi ) ( 2 pi I)/(3a) hat(k)`
` vec(B) = vec(B)_(1) + vec(B)_(2) = ( mu_(0))/( 4 pi) (I)/(a) [ (2)/(3) - 2 sqrt(3)]hat (k) = (- mu_(0))/( 4 pi ) ( 2 I)/(a) (1.4) hat ((k)),`
` vec(v) = vcos 60 hat(i) + v sin 60 hat(j) `
` vec(F) = Q (vec(V)xx vec(B)) = Q[ (v)/(2)hat(i) + (sqrt(3))/(2) v hat(j)] xx [ ( mu_(0))/( 4 pi) ( 2.8 I)/(a) hat (K)]`
Now apply ` vec(a) = (vec(F))/(m)`
(b) KEY CONCEPT :The torque acting on the loop in the magnetic field is given by
`vec(tau) = vec(M)xx vec(B)` where `M = IA`
` A = ( area of PMQNP) - ( area of triangle PMN)`
` = 1/3 ( pi a^(2)) - 1/2 xx MNxxPS`
` = (pi a^(2))/(3) - (1)/(2)xx sqrt( 3a)xx (a)/(2) = a^(2)[(pi)/(3) - sqrt(3)/(4)]`
`vec(A) = a^(2)[(pi)/(3) - (sqrt(3))/(4)] hat(k)`
` :. vec(tau) = Ia^(2)[(pi)/(3) - (sqrt(3))/(4)] hat(k)xx hat(i)B`
`vec(tau) = BIa^(2) ((pi)/(3) - (sqrt(3))/(4))hat(j) = 0.614BIa^(2)hat(j)`
The force acting on the loop is zero.