Question

Four long, straight wires, each carrying a current of 5.0 A, are placed in a plane as shown in figure. The points of intersection form a square of side 5.0 cm.
(a) Find the magnetic field at the centre P of the square.
(b) Q₁, Q₂, Q₃, and Q₄, are points situated on the diagonals of the square and at a distance from P that is equal to the diagonal of the square. Find the magnetic fields at these points.

Answer 1

Given:

Let the horizontal wires placed at the bottom and top are denoted as W₁ and W₂ respectively.
Let the vertical wires placed at the right and left to point P are denoted as W₃ and W₄ respectively.
Magnitude of current, I = 5 A
(a) Consider point P.
Magnetic fields due to wires W₁ and W₂ are the same in magnitude, but they are opposite in direction.
Magnetic fields due to wires W₃ and W₄ are the same in magnitude, but they are opposite in direction.
Hence, the net magnetic field is zero.
Net magnetic field at P due to these four wires = 0

(b) Consider point Q₁.
Due to wire W₁, separation of point Q₁ from the wire (d) is 7.5 cm.
So, the magnetic field due to current in the wire is given by
$B_{W_1}=\frac{{\mu }_{0}I}{2\pi d}$
= 4 × 10⁻⁵ T (In upward direction)

Due to wire W₂, separation of point Q₁ from the wire (d) is 2.5 cm.
So, the magnetic field due to current in the wire is given by
$B_{{\mathrm{W}}_2}=\frac{4}{3}\times {10}^{-5}\mathrm{T}$ (In upward direction)

Due to wire W₃, separation of point Q₁ from the wire (d) is 7.5 cm.
So, the magnetic field due to current in the wire is given by
B_W3 = 4 × 10⁻⁵ T (In upward direction)

Due to wire W₄, separation of point Q₁ from the wire (d) is 2.5 cm.
So, the magnetic field due to current in the wire is given by
$B_{{\mathrm{W}}_4}=\frac{4}{3}\times {10}^{-5}\mathrm{T}$ (In upward direction)

∴ Net magnetic field at point Q₁
$$\begin{gathered} B_{{\mathrm{Q}}_1}=\left(4+\frac{4}{3}+4+\frac{4}{3}\right)\times {10}^{-5} \\ =\frac{32}{3}\times {10}^{-5} \\ =1.06\times {10}^{-4}\mathrm{T}(\mathrm{In}\mathrm{upward}\mathrm{direction}) \end{gathered}$$

At point Q₂,
Magnetic field due to wire W₁:
B_W1 = 4 × 10⁻⁵ T (In upward direction)
Magnetic field due to wire W₂:
$B_{{\mathrm{W}}_2}=\frac{4}{3}\times {10}^{-5}\mathrm{T}$ (In upward direction)
Magnetic field due to wire W₃:
$B_{{\mathrm{W}}_3}=\frac{4}{3}\times {10}^{-5}\mathrm{T}$ (In downward direction)
Magnetic field due to wire W₄:
${\mathrm{B}}_{{\mathrm{W}}_4}=4\times {10}^{-5}\mathrm{T}$ (In downward direction)

∴ Net magnetic field at point Q_{2,
$B_{Q_2}=0$}
Similarly, at point Q₃, the magnetic field is 1.1 × 10⁻⁴ T (in downward direction) and at point Q₄, the magnetic field is zero.