A conducting circular loop is placed in a uniform magnetic field $0.04\, T$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at $2\, mm/s.$ The induced emf in the loop when the radius is $2\, cm$ is

Question

Which of the following correctly represents the AND logic gate?

Answer 1

We are given a conducting circular loop placed perpendicular to a uniform magnetic field of 0.04\, T. The radius of the loop, which is initially 2\, cm, starts shrinking at a rate of 2\, mm/s. Our task is to find the induced emf in the loop at that instant.

The formulae we'll use involve the rate of change of magnetic flux, which is given by Faraday's law of electromagnetic induction:

\text{EMF} = -\frac{{d\Phi}}{{dt}}

Here, \Phi is the magnetic flux through the loop.

Step 1: Calculate the magnetic flux $\Phi$:

The formula for the magnetic flux is:

\Phi = B \cdot A

Where B is the magnetic field and A is the area of the loop. The area of a circle is given by:

A = \pi r^2

Substituting the area formula into the flux equation gives:

\Phi = B \cdot \pi r^2

Step 2: Differentiate with respect to time to find the rate of change of flux:

The rate of change of flux is:

\frac{{d\Phi}}{{dt}} = B \cdot \pi \cdot \frac{{d}}{{dt}}(r^2)

Using the chain rule, we have:

\frac{{d}}{{dt}}(r^2) = 2r \cdot \frac{{dr}}{{dt}}

Putting this into the equation for \frac{{d\Phi}}{{dt}} gives:

\frac{{d\Phi}}{{dt}} = B \cdot \pi \cdot 2r \cdot \frac{{dr}}{{dt}}

Step 3: Substitute the given values:

B = 0.04 T
r = 2 cm = 0.02 m
$\frac{{dr}}{{dt}}$ = -2 mm/s = -0.002 m/s (negative sign because the radius is shrinking)

Substitute these values into the differentiated flux equation:

\frac{{d\Phi}}{{dt}} = 0.04 \cdot \pi \cdot 2 \cdot 0.02 \cdot (-0.002)

\frac{{d\Phi}}{{dt}} = -3.2 \cdot 10^{-6} \pi

Thus, the induced EMF is:

\text{EMF} = -\frac{{d\Phi}}{{dt}} = 3.2 \times 10^{-6} \pi\, V

Converting to microvolts:

3.2 \pi \mu V

The correct option is therefore 3.2\pi\mu V.

Answer 2