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

\[B_{W_2}  = \frac{4}{3} \times  {10}^{- 5}   T\]    (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

\[B_{W_4}  = \frac{4}{3} \times  {10}^{- 5}   T\]   (In upward direction)

∴ Net magnetic field at point Q₁

\[B_{Q_1}  = \left( 4 + \frac{4}{3} + 4 + \frac{4}{3} \right) \times  {10}^{- 5}   \] 

\[             = \frac{32}{3} \times  {10}^{- 5} \] 

\[             = 1 . 06 \times  {10}^{- 4}   T        (\text{ In  upward  direction })\]

At point Q₂,
Magnetic field due to wire W₁:
B_W1 = 4 × 10⁻⁵ T   (In upward direction)
Magnetic field due to wire W₂:

\[B_{W_2}  = \frac{4}{3} \times  {10}^{- 5}   T\]    (In upward direction)

Magnetic field due to wire W₃:

\[B_{W_3}  = \frac{4}{3} \times  {10}^{- 5}   T\]   (In downward direction)

Magnetic field due to wire W₄:

\[B_{W_4}  = 4 \times  {10}^{- 5}   T\]      (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.