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Question: If a coil of \(40\) turns and area \(4.0\,c{m^2}\) is suddenly removed from a magnetic field, it is ...

If a coil of 4040 turns and area 4.0 cm24.0\,c{m^2} is suddenly removed from a magnetic field, it is observed that a charge of 2.0×10−4C2.0 \times {10^{ - 4}}C flows into the coil. If the resistance of the coil is 800Ω800\Omega , the magnetic flux density in Wb m−2Wb{\text{ }}{m^{ - 2}} is:
A. 0.50.5
B. 1.01.0
C. 1.51.5
D. 2.02.0

Explanation

Solution

In this question, we will be using the concept of Faraday's law of induced emf and remember that the induced emf also depends on the number of turns in the coil. Plugging in the values into an established relation obtained from the preceding statements should lead you to the correct result.

Formula used:
Q=N△ϕRQ = N\dfrac{{\vartriangle \phi }}{R}
Where, NN is the number of turns, △ϕ\vartriangle \phi is the change in magnetic flux and RR is the resistance.

Complete step by step answer:
As we know that Q=N△ϕRQ = N\dfrac{{\vartriangle \phi }}{R}
Where, △ϕ\vartriangle \phi denotes change in the magnetic flux.
Also, we know that the magnetic flux is the product of magnetic field and cross sectional area i.e., ϕ=BA\phi = BA.
Therefore, we can say that,
Q=N△ϕRQ = N\dfrac{{\vartriangle \phi }}{R}
Now substituting the value of ϕ\phi in above equation we get,
Q=N△ϕR ⇒Q=NBARQ = N\dfrac{{\vartriangle \phi }}{R} \\\ \Rightarrow Q = \dfrac{{NBA}}{R}
Where NN is the number of turns.

So, B=QRNAB = \dfrac{{QR}}{{NA}}
And is given that,
Q=2×10−4CQ = 2 \times {10^{ - 4}}C
Resistance, R=80ΩR = 80\Omega and Cross sectional area, A=4 cm2=4×10−4m2A = 4\,c{m^2} = 4 \times {10^{ - 4}}{m^2}.
Now, putting all the values in the obtained formula,
B=QRNA ⇒B=2×10−4×8040×4×10−4 ⇒B=2×8040×4 ∴B=1 Wb m−2B = \dfrac{{QR}}{{NA}} \\\ \Rightarrow B = \dfrac{{2 \times {{10}^{ - 4}} \times 80}}{{40 \times 4 \times {{10}^{ - 4}}}} \\\ \Rightarrow B = \dfrac{{2 \times 80}}{{40 \times 4}} \\\ \therefore B = 1\,Wb{\text{ }}{m^{ - 2}}
So, the magnetic flux density is 1 Wb m−21\,Wb{\text{ }}{m^{ - 2}} .

Hence the correct option is B.

Note: Recall that the other ways to change the emf induced in a coil in addition to reversing the direction of the applied field are by:
-Establishing relative motion between the field and the coil.
-Changing the number of turns in the coil
-By changing the strength of the magnetic field itself, which in turn changes the magnetic flux and consequently, the emf.