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Step 1: Identify the Known Quantities
• Magnetic field, $B = 0.04 \,\text{T}$
• Initial radius of the loop at the instant considered, $r = 2 \,\text{cm} = 2 \times 10^{-2}\,\text{m}$
• Rate of change of radius, $\frac{dr}{dt} = 2 \times 10^{-3}\,\text{m/s}$
Step 2: Write the Expression for Magnetic Flux
The magnetic flux $\phi$ through the circular loop when its plane is perpendicular to the magnetic field is given by:
$\phi = B \times \text{(area)} = B \pi r^2$
Step 3: Find the Induced EMF
According to Faraday’s law of electromagnetic induction, the induced EMF $e$ is given by the time rate of change of magnetic flux:
$|e| = \left|\frac{d\phi}{dt}\right| = \left|\frac{d}{dt}\bigl(B \pi r^2\bigr)\right|$
Since $B$ and $\pi$ are constants, we have:
$|e| = B \pi \cdot 2 r \frac{dr}{dt}$
Step 4: Substitute Numerical Values
Substitute $B = 0.04\,\text{T}$, $r = 2 \times 10^{-2}\,\text{m}$, and $\frac{dr}{dt} = 2 \times 10^{-3}\,\text{m/s}$:
$|e| = 0.04 \times \pi \times 2 \times \bigl(2 \times 10^{-2}\bigr) \times \bigl(2 \times 10^{-3}\bigr)$
Perform the multiplication step by step:
Inside the parentheses: $2 \times 10^{-2} \times 2 \times 10^{-3} = 4 \times 10^{-5}$
Now multiply by the remaining factors: $0.04 \times \pi \times 2 \times (4 \times 10^{-5})$
$0.04 \times 2 = 0.08;$ then $0.08 \times 4 \times 10^{-5} = 0.32 \times 10^{-5} = 3.2 \times 10^{-6}$
Thus,
$|e| = 3.2 \times 10^{-6}\,\text{V} \times \pi = 3.2\,\pi \times 10^{-6}\,\text{V}$
We can write $10^{-6}\,\text{V}$ as $\mu\text{V}$ (microvolt):
$\boxed{3.2\,\pi\,\mu\text{V}}$
