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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.
Solution
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 \[\frac{\mu_0 i}{4\pi d}\] .
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