An emf is induced in a conducting loop of wire 1.34 m long as its shape is changed from square to circular.
Find the average magnitude of the induced emf if the change in shape occurs in 0.155 s and the local 0.457-T magnetic field is perpendicular to the plane of the loop.

We can solve this problem using **Faraday's Law of Electromagnetic Induction**. Faraday’s Law states that the induced electromotive force (emf) in a closed loop is proportional to the rate of change of magnetic flux through the loop.

The formula for the induced emf is:

\[
\text{emf} = \left|\frac{\Delta \Phi_B}{\Delta t}\right|
\]

where:
- \(\Phi_B = B \cdot A\) is the magnetic flux, \(B\) is the magnetic field, and \(A\) is the area of the loop.
- \(\Delta \Phi_B\) is the change in magnetic flux.
- \(\Delta t\) is the time interval over which the change occurs.

### Given dataWe can solve this problem using **Faraday's Law of Electromagnetic Induction**. Faraday’s Law states that the induced electromotive force (emf) in a closed loop is proportional to the rate of change of magnetic flux through the loop.

The formula for the induced emf is:

\[
\text{emf} = \left|\frac{\Delta \Phi_B}{\Delta t}\right|
\]

where:
- \(\Phi_B = B \cdot A\) is the magnetic flux, \(B\) is the magnetic field, and \(A\) is the area of the loop.
- \(\Delta \Phi_B\) is the change in magnetic flux.
- \(\Delta t\) is the time interval over which the change occurs.

### Given 
- Length of wire \(L = 1.34 \, \text{m}\)
- Magnetic field \(B = 0.457 \, \text{T}\)
- Time interval \(\Delta t = 0.155 \, \text{s}\)

### Step 1: Calculate the area of the square loop
For a square loop, the perimeter is equal to the length of the wire. The perimeter of the square is \(4s\), where \(s\) is the length of one side of the square.

\[
4s = L \implies s = \frac{L}{4} = \frac{1.34 \, \text{m}}{4} = 0.335 \, \text{m}
\]

The area of the square \(A_{\text{square}}\) is:

\[
A_{\text{square}} = s^2 = (0.335 \, \text{m})^2 = 0.112225 \, \text{m}^2
\]

### Step 2: Calculate the area of the circular loop
The same wire forms a circular loop with circumference \(C = L\). The circumference of a circle is \(2\pi r\), where \(r\) is the radius of the circle.

\[
2\pi r = L \implies r = \frac{L}{2\pi} = \frac{1.34 \, \text{m}}{2\pi} \approx 0.213 \, \text{m}
\]

The area of the circular loop \(A_{\text{circle}}\) is:

\[
A_{\text{circle}} = \pi r^2 = \pi (0.213 \, \text{m})^2 \approx 0.1425 \, \text{m}^2
\]

### Step 3: Calculate the change in area
\[
\Delta A = A_{\text{circle}} - A_{\text{square}} = 0.1425 \, \text{m}^2 - 0.112225 \, \text{m}^2 = 0.030275 \, \text{m}^2
\]

### Step 4: Calculate the change in magnetic flux
Since the magnetic field is perpendicular to the loop, the change in magnetic flux is:

\[
\Delta \Phi_B = B \cdot \Delta A = 0.457 \, \text{T} \cdot 0.030275 \, \text{m}^2 \approx 0.01384 \, \text{Wb}
\]

### Step 5: Calculate the induced emf
Now, using Faraday's Law:

\[
\text{emf} = \left|\frac{\Delta \Phi_B}{\Delta t}\right| = \frac{0.01384 \, \text{Wb}}{0.155 \, \text{s}} \approx 0.0893 \, \text{V}
\]

### Final Answer:
The average magnitude of the induced emf is approximately **0.0893 V**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the induced emf change if the magnetic field were doubled?
2. What would happen to the emf if the shape change took twice as long?
3. How does the direction of the magnetic field affect the induced emf?
4. How would the emf change if the loop remained square but the magnetic field strength changed?
5. Can the induced emf be zero under certain conditions?

**Tip**: The faster the change in magnetic flux (smaller \(\Delta t\)), the larger the induced emf will be.

An emf is induced in a conducting loop of wire 1.34 m long as its shape is changed from square to circular. Find the average magnitude of the induced emf if the change in shape occurs in 0.155 s and the local 0.457-T magnetic field is perpendicular to the plane of the loop.

Learn how to calculate the average induced emf in a wire loop as it changes from square to circular. This problem involves Faraday's Law, magnetic flux, and geometry. Data: wire length = 1.34 m, magnetic field = 0.457 T, time interval = 0.155 s.

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