A long circular tube of length 10 m and radius 0.3 m carries a current along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current va

Question

A long circular tube of length 10 m and radius 0.3 m carries a current $I$ along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $I = I_{0} \cos (300 t)$ where $I_{0}$ is constant. If the magnetic moment of the loop is $N μ_{0} I_{0} \sin (300 t),$ then $' N'$ is ________. [2011]

Smart Achievers · Accepted Answer

(6)

Consider the circular tube as a long solenoid. The wires are closely wound. Magnetic field B inside the solenoid $I = I_{0} \cos (300 t)$

$∴$ $B = \frac{μ_{0} I}{L}$ $[∵ n i = \frac{I}{L}]$

Flux passing through the circular coil, $ϕ = B A$ $= (\frac{μ_{0} I}{L}) (π r^{2})$

Induced emf, $e = - \frac{d ϕ}{d t} = - (\frac{μ_{0} π r^{2}}{L}) \frac{d I}{d t}$

Induced current, $i = \frac{e}{R} = - (\frac{μ_{0} π r^{2}}{L R}) \frac{d I}{d t}$

Magnetic moment, $M = i A = i π r^{2}$

or $M = - (\frac{μ_{0} π^{2} r^{4}}{L R}) \frac{d I}{d t}$ ...(i)

Given, $I = I_{0} \cos (300 t)$

$∴$ $\frac{d I}{d t} = - 300 I_{0} \sin (300 t)$

$M = (\frac{300 π^{2} r^{4}}{L R}) μ_{0} I_{0} \sin (300 t)$

$M = N μ_{0} I_{0} \sin (300 t)$

$N = \frac{300 π^{2} r^{4}}{L R}$

$N = \frac{300 {(\frac{22}{7})}^{2} {(0.1)}^{4}}{(10) (0.005)}$

$N = 5.926 \approx 6$

Solution

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