Electromagnetic Induction and Alternating Current 1 Question 1

####1. A very long solenoid of radius R is carrying current I(t)=kteαt(k>0), as a function of time (t0). Counter clockwise current is taken to be positive. A circular conducting coil of radius 2R is placed in the equatorial plane of the solenoid and concentric with the solenoid. The current induced in the outer coil is correctly depicted, as a function of time, by

(Main 2019, 9 April II)

(a)

alt text

(b)

(c)

(d)

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Answer:

Correct Answer: 1. (d)

Solution:

  1. Magnetic flux associated with the outer coil is

φouter =μ0πNRI=μ0NπR(kteαt)=Cteαt

where,

C=μ0NπRk=constant

Induced emf,

e=dφouter dt=Ceαt+(αCteαt)=Ceαt(1αt)

Induced current, I=e Resistance 

 At t=0,I=ve

The correct graph representing this condition is given in option (d).

Alternate Solution

Given solenoid is shown below as,

At t=0, current in solenoid

=I(t=0)=k(0)eα0=0

Graph of eαt and kt versus time can be shown as,

 As, I=kteαt Initially, kt>eαt

So, current is increasing in magnitude.

Finally, after a short time kt<eαt. So, current is decreasing in magnitude.

But in both cases, it remains positive or counter clockwise. So, current induced is at first anti-clockwise (following Lenz’s law) and then it becomes clockwise and finally reduces to zero as t.

So, correct graph of induced current is



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