Question

Figure shows a long wire bent at the middle to form a right angle. Show that the magnitudes of the magnetic fields at the point P, Q, R and S are equal and find this magnitude. 

Answer 1

As shown in the figure, points P, Q, R and S lie on a circle of radius d.
Let the wires be named W₁ and W₂.  

Now,
At point P, the magnetic field due to wire W₁ is given by
B₁ = 0
At point P, the magnetic field due to wire W₂ is given by

\[B_2  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in outward direction)

\[\Rightarrow  B_{net}  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in outward direction)

At point Q, the magnetic field due to wire W₁ is given by

\[B_1  = \frac{\mu_0 i}{4\pi d}\] (Perpendicular to the plane in inward direction)
At point Q, the magnetic field due to wire W₂ is given by
B₂ = 0

\[\Rightarrow  B_{net}  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in inward direction)

At point R, the magnetic field due to wire W₁ is given by
B₁ = 0
At point R, the magnetic field due to wire W₂ is given by 

\[\Rightarrow  B_{net}  = \frac{\mu_0 i}{4\pi d}\] (Perpendicular to the plane in inward direction)

\[\Rightarrow  B_{net}  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in inward direction)

At point S, the magnetic field due to wire W₁ is given by 

\[B_1  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in outward direction) 

At point S, the magnetic field due to wire W₂ is given by
B₂ = 0

\[\Rightarrow  B_{net}  = \frac{\mu_0 i}{4\pi d}\]  (Perpendicular to the plane in outward direction)

Hence, the magnitude of the magnetic field at points P, Q,  R and S is \[\frac{\mu_0 i}{4\pi d}\] .