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MCQEasyJEE 2023Faraday's Laws of EMI

JEE Physics 2023 Question with Solution

The magnetic field BB crossing normally a square metallic plate of area 4 m24 \, \text{m}^2 is changing with time as shown in the figure. The magnitude of induced emf in the plate during t=2 st = 2 \, \text{s} to t=4 st = 4 \, \text{s} is _____ mV.

  • A

    8 V8 \, \text{V}

  • B

    12 V12 \, \text{V}

  • C

    16 V16 \, \text{V}

  • D

    10 V10 \, \text{V}

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The area of the square metallic plate is A=4 m2A = 4 \, \text{m}^2. The magnetic field changes from B1=4 TB_1 = 4 \, \text{T} at t=2 st = 2 \, \text{s} to B2=8 TB_2 = 8 \, \text{T} at t=4 st = 4 \, \text{s}.

Find: The magnitude of the induced emf in the plate during this interval.

Using Faraday's law of induction:

emf=−dΦdt\text{emf} = -\frac{d\Phi}{dt}

The magnetic flux is:

Φ=B×A\Phi = B \times A

Since the field crosses the plate normally and the area is constant,

∣emf∣=A∣dBdt∣|\text{emf}| = A\left|\frac{dB}{dt}\right|

From the graph data used in the solution,

ΔB=B2−B1=8−4=4 T\Delta B = B_2 - B_1 = 8 - 4 = 4 \, \text{T} Δt=4−2=2 s\Delta t = 4 - 2 = 2 \, \text{s}

So,

dBdt=4 T2 s=2 T/s\frac{dB}{dt} = \frac{4 \, \text{T}}{2 \, \text{s}} = 2 \, \text{T/s}

Therefore,

∣emf∣=4 m2×2 T/s=8 V|\text{emf}| = 4 \, \text{m}^2 \times 2 \, \text{T/s} = 8 \, \text{V}

Thus, the magnitude of the induced emf is 8 V8 \, \text{V}. The correct option is A.

Common mistakes

  • Using the magnetic field value directly instead of its rate of change is incorrect because Faraday's law depends on dΦdt\frac{d\Phi}{dt}, not on Φ\Phi alone. First find the slope of the BB-versus-tt graph, then multiply by area.

  • Ignoring the plate area gives a wrong emf because magnetic flux is Φ=BA\Phi = BA for normal incidence. After finding dBdt\frac{dB}{dt}, multiply by A=4 m2A = 4 \, \text{m}^2.

  • Confusing sign with magnitude is a conceptual error. The negative sign in Faraday's law gives the direction of induced emf, but the question asks for the magnitude. Therefore use the absolute value in the final step.

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