Four long wires each carrying current I as shown in the are placed at the points `A,B,C` and `D` Find the magnitude and direction of
(i) magnetic field the centre of the square
(ii) force per metre acting on wire at point `D`
.

Answer

(a)`mu_(0)/(4pi)((4I)/a)` along `Y`-axis ,(b)`(mu_(0)I^(2)(-hatj-3hati))/(8pia)`

Explanation

(a)`B=4B_(1)cos45^(@)`
`=4xxmu_(0)/(2pi),I/(asqrt2)xx1/sqrt2`
(b)Magnetic field at point `D`
`vecB_(A)=mu_(0)/(2pi)(I/(2a))(hatj)`
`vecB_(B)=mu_(0)/(2pi)(I/(2sqrt2a))((hati+hatj)/sqrt2)`
`vecB_(C)=mu_(0)/(2pi)(I/(2a))(hat(-i))`
Net magnetic field at `D`:
`vecB=B_(A)+vecB_(B)+vecB_(C)=(mu_(0)I)/(4pia)(hati)/2-(3hatj)/2)`
Force per meter acting on wire at point `D`
`vecF=I[(+hatk)xxvecB]=(mu_(0)I^(2))/(8pia)(-3hati-hatj)`