Q: Two circular coils made of same material having radii 20 cm & 30 cm have turns 100 & 50 respectively. If they are connected (a) in series (b) in parallel (c) separately across a source of emf find the ratio of magnetic inductions at the centre of circles in each case

Sol: (a) As $\large B = \frac{\mu_0 n i}{2 r}$

If coils are in series then i is same in both .

$\large B \propto \frac{n}{r} $

$\large \frac{B_1}{B_2} = \frac{100}{50} \times \frac{30}{20} = 3 : 1 $

(b)If coils are parallel then potential difference is same .

$\large i \propto \frac{1}{R} $ ; Where $\large R = \rho \frac{2\pi n r}{A} $ ; where A is area of cross section of wire which is same for both .

$\large R \propto n r \; , i \propto \frac{1}{n r} $

$\large B \propto \frac{n}{r} \times \frac{1}{n r} $

$\large B \propto \frac{1}{r^2} $

$\large \frac{B_1}{B_2} = (\frac{30}{20})^2 = \frac{9}{4} $

(c) For the coils, potential difference is same

$\large i \propto \frac{1}{R} \; , R \propto n r$

$\large i \propto \frac{1}{ n r} $

$\large B \propto \frac{1}{r^2} $

$\large \frac{B_1}{B_2} = \frac{9}{4} $