SATHEE: Electromagnetic Induction and Alternating Current 7 Question 25

Electromagnetic Induction and Alternating Current 7 Question 25

28. A long solenoid of radius $a$ and number of turns per unit length $n$ is enclosed by cylindrical shell of radius $R$ , thickness $d (d ≪ R)$ and length $L$ .

A variable current $i = i_{0} \sin ω t$ flows through the solenoid. If the resistivity of the material of cylindrical shell is $ρ$ , find the induced current in the shell.

$(2005, 4$ M)

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Answer:

Correct Answer: 28. $i = \frac{μ_{0} L d n a^{2} I_{0} ω \cos ω t}{2 ρ R}$

Solution:

Outside the solenoid, net magnetic field is zero. It can be assumed only inside the solenoid and equal to $μ_{0} n I$ .

Induced $e = - \frac{d φ}{d t} = - \frac{d}{d t} (μ_{0} n I π a^{2})$

or $| e | = (μ_{0} n π a^{2}) (I_{0} ω \cos ω t)$

Resistance of the cylindrical vessel, $R = \frac{ρ l}{s} = \frac{ρ (2 π R)}{L d}$

$∴$ Induced current $i = \frac{| e |}{R} = \frac{μ_{0} L d n a^{2} I_{0} ω \cos ω t}{2 ρ R}$

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Electromagnetic Induction and Alternating Current 7 Question 25

28. A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d(d≪R) and length L.

Answer:

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28. A long solenoid of radius $a$ and number of turns per unit length $n$ is enclosed by cylindrical shell of radius $R$ , thickness $d (d ≪ R)$ and length $L$ .