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Three long straight wires in the XZ-plane, each carrying I, cross at the origin of coordinates, as shown in the figure below. Let x, y and z denote the unit vectors in the x, y and z-directions respectively. The magnetic field as a function of x, with y=0 and z=0, is:
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(A) B=3μ0I2πxx
(B) B=3μ0I2πxy
(C) B=μ0I2πx(1+22)y
(D) B=μ0I2πx(1+22)(y)

Answer
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Hint To solve this question, we have to use the formula for the magnetic field due to an infinitely long carrying wire. And for getting the direction, we need to use the right hand thumb rule.
The formula which is used in solving this question is given by
B=μ0I2πr , here B is the magnetic field produced by an infinitely long wire carrying a current I at a perpendicular distance of r from it.

Complete step by step answer
Let us consider a point P on the x-axis at a distance of x from the origin, as shown in the following figure.

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We know that the magnetic field due to an infinitely long wire is given by
B=μ0I2πr (1)
So we need to calculate the perpendicular distance of the point P from each of the wires A, B and C.
For wire A:
Consider the following figure.
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From the figure, we see that the perpendicular distance of the point P is
r1=PM
In the triangle OPM, we have
PM=OPcos45
r1=xcos45
So we get
r1=x2 (2)
So from (1) the magnetic field due to wire A at the point P is
BA=μ0I2πr1
From (2)
BA=μ0I2πx2
BA=μ0I2πx
By the right hand thumb rule, the direction of this field is along the positive y-axis. So in terms of the unit vector we have
BA=μ0I2πxy
Similarly, the magnetic field due to the wire B at the point P
BB=μ0I2πxy
Also, the magnetic field at P due to wire C is
BC=μ0I2πxy
So the net magnetic field at point P is
B=BA+BB+BC
From (3) (4) and (5)
B=μ0I2πxy+μ0I2πxy+μ0I2πxy
On simplifying we get
B=μ0I2πx(1+22)y
Thus, this is the required magnetic field as a function of x.
Hence, the correct answer is option (C).

Note
The given set of axes is not the standard set of coordinate axes. So, for getting the direction of the positive y-axis, we use the right hand thumb rule. We should not assume the direction by ourselves.