Figure shows a long wire bent at the middle to form a right angle. Show that the magnitudes of the magnetic fields at the points $Q$ and $R$ are unequal and find these magnitudes. The wire $w_{1}$ and the circumference circle are coplanar and $w_{2}$ is perpendicular to plane of paper. Also find the ratio of field at $Q$ and $R$ .

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Solution

Given $w_{2} ⊥ r$ and $w_{1}$ co-planner with circle.
$B_{Q} =$ magnetic field at $Q$ due to wire $(w_{1} + w_{2})$ where $B$ due to $w_{!} = 0$ since $θ = 0 °$
$⟹ B_{Q}$ magnetic field due to $w_{2} = \frac{μ_{0} I}{4 π d}$
$B_{R} =$ Magnetic field at $R$ due to wire $(w_{1} + w_{2})$
$= \frac{μ_{0} I}{4 π d} + \frac{μ_{0} I}{4 π d} = \frac{μ_{0} I}{2 π d}$
$B_{R} = 2 B_{Q}$

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Given w2⊥r and w1 co-planner with circle.BQ= magnetic field at Q due to wire (w1+w2) where B due to w!=0 since θ=0°⟹BQ magnetic field due to w2=μ0I4πdBR= Magnetic field at R due to wire (w1+w2)=μ0I4πd+μ0I4πd=μ0I2πdBR=2BQ