We can solve this problem using Faraday's Law of Induction, which states that the induced electromotive force (emf) in a loop is given by:

$\mathcal{E} = -\frac{d\Phi_B}{dt}$

Where:

• $\mathcal{E}$ is the induced emf,
• $\Phi_B$ is the magnetic flux, and
• $\frac{d\Phi_B}{dt}$ is the rate of change of the magnetic flux.

The magnetic flux $\Phi_B$ through the loop is given by:

$\Phi_B = B \cdot A$

Where:

• $B$ is the magnetic field strength,
• $A$ is the area of the loop.

Now, the area of the circular loop is related to its circumference. If $C$ is the circumference, then the radius $r$ of the loop is:

$C = 2\pi r$

Thus, the radius of the loop is:

$r = \frac{C}{2\pi}$

The area $A$ of the loop is then:

$A = \pi r^2 = \pi \left(\frac{C}{2\pi}\right)^2 = \frac{C^2}{4\pi}$

Thus, the magnetic flux is:

$\Phi_B = B \cdot \frac{C^2}{4\pi}$

To find the induced emf, we differentiate the flux with respect to time:

$\frac{d\Phi_B}{dt} = B \cdot \frac{d}{dt} \left( \frac{C^2}{4\pi} \right)$

Using the chain rule:

$\frac{d\Phi_B}{dt} = B \cdot \frac{1}{4\pi} \cdot 2C \cdot \frac{dC}{dt}$

Simplifying:

$\frac{d\Phi_B}{dt} = \frac{B C}{2\pi} \cdot \frac{dC}{dt}$

Now, we can substitute the given values:

• The magnetic field $B = 0.500 \, \text{T}$,
• The rate of change of the circumference $\frac{dC}{dt} = -11.0 \, \text{cm/s} = -0.11 \, \text{m/s}$,
• The initial circumference $C = 164 \, \text{cm} = 1.64 \, \text{m}$.

Thus, the rate of change of magnetic flux becomes:

$\frac{d\Phi_B}{dt} = \frac{0.500 \times 1.64}{2\pi} \times (-0.11)$

Now, calculating this expression:

$\frac{d\Phi_B}{dt} = \frac{0.820}{2\pi} \times (-0.11) \approx -0.0144 \, \text{Wb/s}$

Finally, the induced emf is:

$\mathcal{E} = -\frac{d\Phi_B}{dt} = 0.0144 \, \text{V}$

Thus, the magnitude of the induced emf is:

$\boxed{0.0144 \, \text{V}}$

Let me know if you need further details or clarifications!

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Related Questions:

1. What is the relationship between magnetic flux and the area of a loop?
2. How does the rate of change in the magnetic flux affect the induced emf?
3. What is the significance of the negative sign in Faraday’s Law?
4. How can the induced emf be affected by the strength of the magnetic field?
5. What would happen to the induced emf if the magnetic field strength were doubled?

Tip: Always ensure the units are consistent when solving electromagnetism problems—converting all measurements to SI units is crucial!