Answer :
To find the electromotive force (EMF) generated in the coil, we can use Faraday's law of electromagnetic induction. Here is a detailed, step-by-step solution to the problem:
### Step 1: Gather the given information
- Number of loops (N): 450
- Radius of each loop (r): 0.0254 meters
- Initial magnetic field (B₁): 0 Tesla
- Final magnetic field (B₂): 3.00 Tesla
- Time interval (Δt): 1.55 seconds
### Step 2: Calculate the area of one loop
The area [tex]\( A \)[/tex] of a single loop is given by the formula for the area of a circle:
[tex]\[ A = \pi \times r^2 \][/tex]
Substituting the radius given:
[tex]\[ A = \pi \times (0.0254)^2 \][/tex]
### Step 3: Calculate the change in magnetic flux
Magnetic flux ([tex]\( \Phi \)[/tex]) is the product of the magnetic field and the area it penetrates ([tex]\( \Phi = B \times A \)[/tex]).
The change in magnetic flux ([tex]\( \Delta \Phi \)[/tex]) through one loop when the magnetic field changes from [tex]\( B₁ \)[/tex] to [tex]\( B₂ \)[/tex] is:
[tex]\[ \Delta \Phi = A \times (B₂ - B₁) \][/tex]
[tex]\[ \Delta \Phi = A \times (3.00 - 0) \][/tex]
For all the loops (since there are 450), the total change in magnetic flux ([tex]\( \Delta \Phi_{total} \)[/tex]) is:
[tex]\[ \Delta \Phi_{total} = N \times \Delta \Phi \][/tex]
### Step 4: Calculate the EMF
According to Faraday's law, the induced EMF ([tex]\( \mathcal{E} \)[/tex]) is equal to the negative rate of change of magnetic flux through the coil. For calculation purposes, we are interested in the magnitude:
[tex]\[ \mathcal{E} = \frac{\Delta \Phi_{total}}{\Delta t} \][/tex]
### Solution outcome
By substituting the given values and performing these calculations:
- Total change in magnetic flux through the coil is approximately: 2.7362 Weber
- The EMF generated is approximately: 1.7653 Volts
Therefore, the EMF generated in the coil is approximately 1.77 Volts.
### Step 1: Gather the given information
- Number of loops (N): 450
- Radius of each loop (r): 0.0254 meters
- Initial magnetic field (B₁): 0 Tesla
- Final magnetic field (B₂): 3.00 Tesla
- Time interval (Δt): 1.55 seconds
### Step 2: Calculate the area of one loop
The area [tex]\( A \)[/tex] of a single loop is given by the formula for the area of a circle:
[tex]\[ A = \pi \times r^2 \][/tex]
Substituting the radius given:
[tex]\[ A = \pi \times (0.0254)^2 \][/tex]
### Step 3: Calculate the change in magnetic flux
Magnetic flux ([tex]\( \Phi \)[/tex]) is the product of the magnetic field and the area it penetrates ([tex]\( \Phi = B \times A \)[/tex]).
The change in magnetic flux ([tex]\( \Delta \Phi \)[/tex]) through one loop when the magnetic field changes from [tex]\( B₁ \)[/tex] to [tex]\( B₂ \)[/tex] is:
[tex]\[ \Delta \Phi = A \times (B₂ - B₁) \][/tex]
[tex]\[ \Delta \Phi = A \times (3.00 - 0) \][/tex]
For all the loops (since there are 450), the total change in magnetic flux ([tex]\( \Delta \Phi_{total} \)[/tex]) is:
[tex]\[ \Delta \Phi_{total} = N \times \Delta \Phi \][/tex]
### Step 4: Calculate the EMF
According to Faraday's law, the induced EMF ([tex]\( \mathcal{E} \)[/tex]) is equal to the negative rate of change of magnetic flux through the coil. For calculation purposes, we are interested in the magnitude:
[tex]\[ \mathcal{E} = \frac{\Delta \Phi_{total}}{\Delta t} \][/tex]
### Solution outcome
By substituting the given values and performing these calculations:
- Total change in magnetic flux through the coil is approximately: 2.7362 Weber
- The EMF generated is approximately: 1.7653 Volts
Therefore, the EMF generated in the coil is approximately 1.77 Volts.
