To find the induced electromotive force (emf) in the scenario you've described, we can apply Faraday's law of electromagnetic induction. This law states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Let's break down the problem step by step.
Step 1: Calculate the Initial Magnetic Flux
The initial configuration of the wire is a circular loop. The circumference of the circle is given as 88 cm, which allows us to find the radius and then the area of the circle.
- First, convert the circumference to meters: 88 cm = 0.88 m.
- The formula for the circumference of a circle is \( C = 2\pi r \). Rearranging gives us \( r = \frac{C}{2\pi} = \frac{0.88}{2\pi} \approx 0.14 \, \text{m} \).
- Now, calculate the area \( A \) of the circle using \( A = \pi r^2 \):
\( A \approx \pi (0.14)^2 \approx 0.061 \, \text{m}^2 \).
- The initial magnetic flux \( \Phi_1 \) is given by \( \Phi_1 = B_1 \times A \), where \( B_1 = 2.5 \, \text{Wb/m}^2 \):
\( \Phi_1 = 2.5 \times 0.061 \approx 0.153 \, \text{Wb} \).
Step 2: Calculate the Final Magnetic Flux
Next, we need to find the magnetic flux when the wire is reshaped into a square loop.
- The side length of the square is given as 22 cm, which is 0.22 m.
- The area \( A \) of the square is calculated as \( A = \text{side}^2 = (0.22)^2 = 0.0484 \, \text{m}^2 \).
- The new magnetic flux \( \Phi_2 \) with the increased flux density \( B_2 = 3 \, \text{Wb/m}^2 \) is:
\( \Phi_2 = B_2 \times A = 3 \times 0.0484 \approx 0.1452 \, \text{Wb} \).
Step 3: Determine the Change in Magnetic Flux
The change in magnetic flux \( \Delta \Phi \) is calculated as follows:
- \( \Delta \Phi = \Phi_2 - \Phi_1 = 0.1452 - 0.153 = -0.0078 \, \text{Wb} \).
Step 4: Calculate the Induced EMF
According to Faraday's law, the induced emf \( \mathcal{E} \) can be calculated using the formula:
\( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \), where \( \Delta t = 0.5 \, \text{s} \).
- Substituting the values gives:
\( \mathcal{E} = -\frac{-0.0078}{0.5} = 0.0156 \, \text{V} \) or 15.6 mV.
Final Result
The induced emf in the circuit when the loop is transformed from a circular shape to a square shape and the magnetic flux density is increased is approximately 15.6 mV.
