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`
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Answer

`(mu_(0))/(4pi) ((4I)/(a))` along Y-axis
(ii) `(mu_(0))/(4pi) ((I^(2))/(2a)) sqrt10 , tan^(4)' ((1)/(3)) + pi` with positive axis .

Explanation

`oversetrarrB = oversetrarr(B_(A))=oversetrarr(B_(B))=oversetrarr(B_(C))=oversetrarr(B_(D))`
since `B_(A) =B_(B) =B_(C) =B_(D) = (mu_(0))/(4pi) (2I)/(sqrt2A`
`B =2 sqrt2 B_(A) = 2sqrt2 (mu_(0))/(4pi) (2I)/(sqrt2a)`
`oversetrarrB = (mu_(0))/(4pi) (4I)/(a) (hatj)`
`oversetrarr(F_(D)) = oversetrarr(F_(A)) + oversetrarr(F_(B)) + oversetrarr(F_(C))`
`F_(DX) = F_(Ax) + F_(Bx) + F_(Cx) = - F_(A) - F_(B) cos 45^(@) `
`F_(Dx) = - (mu_(0))/(4pi) (2I^(2))/(2a) - (mu_(0))/(4pi) (2I^(2))/(2sqrt2a) xx (1)/(sqrt2)`
`F_(Dx) = -(mu_(0))/(4pi) (I^(2))/(a) xx (3)/(2)`
`F_(Dy) = F_(Ay) + F_(By) + F_(cy) = F_(B) sin 44^(@) -F_(C)`
`F_(Dy) = -(mu_(0))/(4pi) (I^(2))/(a) xx (1)/(2)`
`F_(D) = sqrt(F_(Dx^(2)) + F_(Dy^(2))) = (mu_(0))/(4pi) (I^(2))/(2a) sqrt10`
Direction ` =pi + tan^(-1) "(F_(y))/(F_(x))=pi + tan^(-1)(1)/(3)`

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