A circular loop is placed in a uniform magnetic field perpendicular to its plane. The field decreases at a constant rate of 0.02 T/s. If the loop's radius is 0.1 m, what is the induced emf in the loop?

8. A circular loop is placed in a uniform magnetic field perpendicular to its plane. The field decreases at a constant rate of \( 0.02 \, \text{T/s} \). If the loop's radius is \( 0.1 \, \text{m} \), what is the induced emf in the loop?

A) \(12.56 \, \text{mV} \)

B) \( 0.628 \, \text{mV} \)

C) \( 0.843 \, \text{mV} \)

D) \( 25.12 \, \text{mV} \)

... Answer is B)

The induced EMF (\( \mathcal{E} \)) in a loop is given by Faraday's law of electromagnetic induction:

\[ \mathcal{E} = \left| \frac{d\Phi}{dt} \right| \]

Magnetic flux (\( \Phi \)) through the loop is given by:

\[ \Phi = B \cdot A \]

Where:

\( B \): Magnetic field strength
\( A = \pi r^2 \): Area of the loop

The rate of change of flux is:

\[ \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} \]

Substitute the given values:

\( r = 0.1 \, \text{m} \): Radius of the loop
\( A = \pi (0.1)^2 = 0.01 \pi \, \text{m}^2 \)
\( \frac{dB}{dt} = -0.02 \, \text{T/s} \): Rate of change of magnetic field

Substitute \( A \) and \( \frac{dB}{dt} \):

\[ \frac{d\Phi}{dt} = 0.01 \pi \cdot 0.02 = 0.0002 \pi \, \text{Wb/s} \]

Now calculate the induced EMF:

\[ \mathcal{E} = \left| \frac{d\Phi}{dt} \right| = 0.0002 \pi \]

Substitute \( \pi \approx 3.14 \):

\[ \mathcal{E} = 0.0002 \cdot 3.14 = 0.000628 \, \text{V} \]

\[ \mathcal{E} = 0.628 \, \text{mV} \]

The induced EMF in the loop is:

\[ \mathcal{E} = 0.628 \, \text{mV} \]

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