Question

In a coil of resistance 100 \[\Omega \], a current is induced by changing a magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is:

                       

A. 275 Wb.
B. 200 Wb.
C. 225 Wb.
D. 250 Wb.

Answer 1

Hint: To solve this particular question, we will use Faraday’s Law of induction also known as Faraday’s first law. We will also use another equation to find electromotive force which includes resistance from the coil. Later, we shall equate the two equations and find the magnitude of change in flux.
Formula used:
 \[\varepsilon =\dfrac{d\phi }{dt}\] Faraday’s law where,
\[\varepsilon \]= electromotive force in volts (V)
\[\phi \] = magnetic flux
t = time in sec (s)
\[\varepsilon =iR\] where,
i = electric current in Amp
R = resistance in ohms (\[\Omega \])

Complete answer:
Faraday’s law of induction is a law of electromagnetism which predicts how a magnetic field will interact with an electric circuit to produce the electromotive force. Faraday’s law basically states that electromotive force (emf) is given by the rate of change of the magnetic flux.

So, from Faraday’s law
\[\varepsilon =\dfrac{d\phi }{dt}\] …………… (1)
Also, induced emf is given by the formula,
\[\varepsilon =iR\] …………..(2)
So, from (1) and (2)
 We can say that,
\[\dfrac{d\phi }{dt}=iR\]
Therefore, to find change in magnetic flux,
\[d\phi =iRdt\]
Now, integrating both sides
We get,
\[\int{d\phi }=R\int{idt}\]….. (\[\because \] R is constant)
Now,
\[\int{idt}\] = area under the given curve (i-t)
\[\therefore \int{idt}=\dfrac{1}{2}\times 10\times 0.5\] ……….. (\[\because \]the area under curve is a triangle)
Therefore,
\[\int{idt}\]= 2.5
Now integrating d\[\phi \] from 0 to \[\phi \] and substituting the value of \[\int{idt}\] and given value of R =100\[\Omega \]
We get,
\[\phi =100\times 2.5\]
Therefore,
\[\phi =250Wb\]

So, the correct answer is “Option D”.

Note:
In the question it is specified that we are to find only the magnitude of the magnetic flux. If we were asked to find the direction as well we can do it by using Lenz’s Law.