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Two circular coils A and B of radii a and b respectively (with b > a ) have their planes perpendicular to the plane of the page. They are separated co-axially by a distance x = b as shown in figure. A transient current I flows through coil B for a very short time interval. If the resistance of coil A is R obtain the expression for the charge that flows through coil A during the short time interval.

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Detailed Solution

Magnetic field at the centre of coil A due to current I in coil B is

$B_{A B} = \frac{μ_{0} I b^{2}}{2 {(b^{2} + x^{2})}^{3 / 2}} = \frac{μ_{0} I}{16 b} (∵ x = \sqrt{3} b)$

Since the magnetic field is along the axis of coil A, it is perpendicular to the plane of A, hence 0 = 0°. Therefore, magnetic flux through A is

$ϕ = B_{A B} \times area of coil A \times \cos 0^{\circ} = \frac{μ_{0} I \times π a^{2}}{16 b}$ $∴ Induced emf is | e | = |\frac{dϕ}{dt}|$

$\Rightarrow I R = \frac{d ϕ}{d t}$

$\Rightarrow I R d t = d ϕ$

$∴ \int Idt = \frac{1}{R} \int dϕ = \frac{ϕ}{R}$

$Q = \frac{ϕ}{R} = \frac{μ_{0} I \times {πa}^{2}}{16 bR}$

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