This problem involves calculating the electromotive force (emf) induced in a shrinking loop of wire due to a changing magnetic flux. Let's break down the steps to solve it.

Given Information:

1. Initial circumference of the loop $C_0 = 169 \, \text{cm} = 1.69 \, \text{m}$.
2. Rate at which the circumference is decreasing $\frac{dC}{dt} = -11.0 \, \text{cm/s} = -0.11 \, \text{m/s}$.
3. The loop is in a uniform magnetic field $B = 0.500 \, \text{T}$, which is perpendicular to the plane of the loop.
4. Time given is $t = 3.00 \, \text{s}$.

To Find:

The magnitude of the induced emf $\mathcal{E}$ after 3.00 seconds.

Key Concept:

The induced emf is given by Faraday's Law of Induction:

$\mathcal{E} = \left| \frac{d\Phi_B}{dt} \right|$

Where $\Phi_B$ is the magnetic flux, which is $\Phi_B = B \cdot A$, with $A$ being the area of the loop.

Since the area depends on the loop's circumference (which is changing over time), we first need to express the area in terms of the circumference.

Step 1: Express Area in Terms of Circumference

For a circular loop, the relationship between the circumference $C$ and the radius $r$ is:

$C = 2 \pi r$

The area of the loop is:

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

Step 2: Differentiate Area with Respect to Time

To find the induced emf, we need the rate of change of the area. Differentiating $A = \frac{C^2}{4\pi}$ with respect to time:

$\frac{dA}{dt} = \frac{2C}{4\pi} \cdot \frac{dC}{dt} = \frac{C}{2\pi} \cdot \frac{dC}{dt}$

Step 3: Calculate the Induced emf

From Faraday’s law, the emf is:

$\mathcal{E} = B \cdot \left| \frac{dA}{dt} \right| = B \cdot \left| \frac{C}{2\pi} \cdot \frac{dC}{dt} \right|$

Now, substitute the known values:

• Initial circumference $C_0 = 1.69 \, \text{m}$
• After 3.00 seconds, the circumference is $C = C_0 + \frac{dC}{dt} \cdot t = 1.69 \, \text{m} + (-0.11 \, \text{m/s}) \cdot 3.00 \, \text{s} = 1.69 \, \text{m} - 0.33 \, \text{m} = 1.36 \, \text{m}$.
• Magnetic field $B = 0.500 \, \text{T}$.

Now calculate:

$\mathcal{E} = 0.500 \, \text{T} \cdot \left| \frac{1.36 \, \text{m}}{2\pi} \cdot (-0.11 \, \text{m/s}) \right|$

Let me calculate the exact result.The magnitude of the induced emf $\mathcal{E}$ after 3.00 seconds is approximately $0.0119 \, \text{V}$.

Would you like more details or explanations on any part of the calculation? Here are some related questions:

1. How is the rate of change of the magnetic flux related to induced emf?
2. What happens if the magnetic field were to change direction?
3. How would the result differ if the magnetic field strength was not constant?
4. What is the significance of the negative sign in $\frac{dC}{dt}$?
5. Can Faraday's Law of Induction be applied to non-circular loops?

Tip: Always check the direction of the magnetic field and area change to apply the right-hand rule for determining the direction of induced current.