
Three long straight wires in the XZ-plane, each carrying , cross at the origin of coordinates, as shown in the figure below. Let , and denote the unit vectors in the , and -directions respectively. The magnetic field as a function of , with and , is:
(A)
(B)
(C)
(D)

Answer
497.1k+ views
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
, here is the magnetic field produced by an infinitely long wire carrying a current at a perpendicular distance of from it.
Complete step by step answer
Let us consider a point P on the x-axis at a distance of from the origin, as shown in the following figure.
We know that the magnetic field due to an infinitely long wire is given by
(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.
From the figure, we see that the perpendicular distance of the point P is
In the triangle , we have
So we get
(2)
So from (1) the magnetic field due to wire A at the point P is
From (2)
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
Similarly, the magnetic field due to the wire B at the point P
Also, the magnetic field at P due to wire C is
So the net magnetic field at point P is
From (3) (4) and (5)
On simplifying we get
Thus, this is the required magnetic field as a function of .
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.
The formula which is used in solving this question is given by
Complete step by step answer
Let us consider a point P on the x-axis at a distance of

We know that the magnetic field due to an infinitely long wire is given by
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.

From the figure, we see that the perpendicular distance of the point P is
In the triangle
So we get
So from (1) the magnetic field due to wire A at the point P is
From (2)
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
Similarly, the magnetic field due to the wire B at the point P
Also, the magnetic field at P due to wire C is
So the net magnetic field at point P is
From (3) (4) and (5)
On simplifying we get
Thus, this is the required magnetic field as a function of
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.
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