Four long wires each carrying current I as shown in Fig. are placed at points A, B, C and D. Find the magnitude and direction of

Force per metre acting on wire at point D

Options

A. `(mu_(0))/(4 pi)((I^2)/(2a)) sqrt(10), pi+tan^(-1)(1/3)` with positive x-axis
B. `(mu_(0))/(4 pi)((I^2)/(2a)) sqrt(10), pi+tan^(-1)(1/3)` with negative x-axis
C. `(mu_(0))/(4 pi)((I^2)/(a)) sqrt(10), pi+tan^(-1)(1/3)` with positive x-axis
D. None of these

Answer

A

Explanation

(i)
`B=B_(1)=B_(2)=B_(3)=B_(4)=(mu_(0)I)/(2 pi sqrt(2)a)`
Net field `=4B cos 45^(@)`
`(4mu_(0)I)/(2 pi sqrt(2)a) (1)/(sqrt(2))=(mu_(0)I)/(pi a)`
(ii)
`F_(1)=(mu_(0))/(2pi) (I^2)/((2a)), F_(2)=(mu_0)/(2pi) (I^2)/(2a)`
`F_(3)=(mu_0)/(2pi) (I^2)/(2sqrt(2)a)`
`F_(x)=-(F_(2)+F_(3) cos 45^(@))=-(mu_(0))/(4 pi) (I^2)/(2a) 3`
`F_(y)=-(F_(1)-F_(3) cos 45^(@))=-(mu_(0))/(4pi)(I^2)/(2a)`
`F_(n et)=sqrt(F_(x)^(2)+F_(y)^(2))= (mu_(0))/(4 pi) (I^2)/(2a) sqrt(10)N//m`
`tan theta =(F_y)/(F_x)=1/3 implies theta=tan^(-1)((1)/(3))`.