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) (QV mu_(0)I)/(m 6a) ((3sqrt3)/(pi)-1),(b) oversetrarrtau =BI((pi)/(3)-(sqrt3)/(4))a^(2)hatj`

Explanation

`B_(1) = (mu_(0))/(4pi) (I)/(a) xx (2pi)/(3) o.`
`B_(2) = (mu_(0))/(4pi) (I)/(x) [sin60^(@) + sin60^(@)]`
`B_(2) = (mu_(0))/(4pi) (I)/(a//2) [2sin60^(@)]=(mu_(0))/(4pi) (4I)/(a) (sqrt3)/(2)ox`
`B = B_(2) -B_(1) =(mu_(0))/(4pi) (I)/(a)[2sqrt3 -(2)/(3)pi]`
`= (mu_(0))/(4pi) xx (2)/(3) (I)/(a) [3sqrt3 -pi]ox`
`F =qVB`
`a = (qVB)/(m) = (qV)/(m) (mu_(0)I)/(6a)[(3sqrt3)/(pi) -1]`
`A =(pia^(2))/(2pi) xx (2pi)/(3) -2 xx (1)/(2) xx (a)/(2) (sqrt3)/(2) a`
`= (pia^(2))/(3) - (sqrt3)/(4) a^(2), oversetrarrA = ((pi)/(3) - (sqrt3)/4)a^(2)hatk`
`vectau = oversetrarrM xx oversetrarrB = I oversetrarrA xx oversetrarrB = I ((pi)/(3)- (sqrt3)/(4))a^(2) hatk xx Bhati`
`=IBa^(2) ((pi)/(3) - (sqrt3)/(4))hatj`
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