A circular plate of radius 0.5 m is placed perpendicular to the direction of the magnetic field of 0.5 Tesla. What is the change of flux in the plate if it rotates and its plane becomes parallel to the direction of the magnetic field?
To calculate the change of flux in the circular plate, we can use the formula:
Change in flux = magnetic field * change in area * cosine(theta)
Where:
- magnetic field is given as 0.5 Tesla
- change in area can be calculated as the difference between the initial and final areas of the circular plate
- cosine(theta) is the angle between the initial and final orientations of the plate, which is 90 degrees in this case since the plate rotates from being perpendicular to parallel to the magnetic field.
1. Calculate the initial area of the circular plate:
Initial area = pi * radius^2
Initial area = pi * (0.5 m)^2
Initial area = 0.7854 m^2
2. Calculate the final area of the circular plate:
Final area = pi * radius^2
Final area = pi * (0.5 m)^2
Final area = 0.7854 m^2
3. Calculate the change in area:
Change in area = Final area - Initial area
Change in area = 0.7854 m^2 - 0.7854 m^2
Change in area = 0 m^2
4. Calculate the change of flux:
Change in flux = magnetic field * change in area * cosine(theta)
Change in flux = 0.5 Tesla * 0 m^2 * cosine(90 degrees)
Change in flux = 0 Nm^2
Therefore, the change of flux in the circular plate is 0 Nm^2.
To determine the change of flux in the circular plate, we need to calculate the initial and final flux and then find the difference between them.
The flux, denoted by Φ, is a measure of the number of magnetic field lines passing through a surface. It is defined as the dot product of the magnetic field vector (B) and the area vector (A) of the surface:
Φ = B ⋅ A
In this scenario, the magnetic field is perpendicular to the circular plate initially (perpendicular to the surface area vector). Therefore, the initial flux is given by:
Φ_initial = B_initial ⋅ A_initial
However, when the plate rotates and becomes parallel to the magnetic field, the angle between the magnetic field vector and the area vector becomes zero. In this case, the final flux can be calculated as:
Φ_final = B_final ⋅ A_final
Since the radius of the circular plate is given as 0.5 meters, the initial area (A_initial) is π * (0.5)^2 = 0.25π square meters, and the final area (A_final) is also 0.25π square meters since the radius doesn't change.
Given that the initial magnetic field (B_initial) is 0.5 Tesla, and the final magnetic field (B_final) is also 0.5 Tesla, we can substitute these values into the flux equations:
Φ_initial = 0.5 Tesla ⋅ 0.25π square meters
Φ_final = 0.5 Tesla ⋅ 0.25π square meters
Simplifying the equations, we find that the initial and final flux are both equal to 0.125π Tesla⋅m^2.
To find the change of flux, we subtract the initial flux from the final flux:
Change of Flux = Φ_final - Φ_initial
= 0.125π Tesla⋅m^2 - 0.125π Tesla⋅m^2
= 0 Tesla⋅m^2
Therefore, the change of flux in the plate as it rotates and becomes parallel to the magnetic field is 0 Tesla⋅m^2.