The 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 of the circle are coplanar and $w_{2}$ is perpendicular to plane of the paper. Also, find the ratio of the field at $Q$ and $R$ .

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Solution

Wire $W_{1}$ is along plane of circus where $W_{2} ⊥^{r}$ to the plane of cicle.
(Refer image)
Let $I$ current flows through the wire
So, at Q magnetic field due to $W_{1}$ is $^{'} 0^{'}$
$(B_{Q})_{d u e t o W_{1}} = 0 (∵ sin θ = 0)$
$(B_{Q})_{d u e t o W_{2}} = 2 (L e t)$
$(B_{R})_{d u e t o W_{1}} = B$ ( $∵$ Same current of same angle makes as $W_{2}$ makes with at point $Q$ )
$(B_{R})_{d u e t o W_{2}} = B$
So, net magnetic field at $^{'} Q^{'} =^{'} B^{'}$
Net magnetic field at $^{'} R^{'} =^{'} 2 B$
So, ration $\frac{B_{Q}}{B_{R}} = \frac{B}{2 B} = \frac{1}{2}$
$B_{R} : B_{R} = 1 : 2$

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