Rotating Loop by 180 Degrees Impacts Magnetic Flux

The magnetic flux through a loop is given by the equation \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(\Phi\) is the magnetic flux, \(B\) is the magnetic field strength, \(A\) is the area of the loop, and \(\theta\) is the angle between the magnetic field and the normal (perpendicular) to the loop. Let's analyze each choice: A. If the original loop is replaced by two loops, each of which has half the area of the original loop, the total area of both loops combined will still be the same as the area of the original loop. Assuming the loops are arranged so that they are still perpendicular to the magnetic field and occupy the same total area, the total magnetic flux will remain unchanged. B. If the loop moves at constant velocity while remaining perpendicular to and within the uniform magnetic field, its area and orientation relative to the magnetic field remain the same. Since neither \(B\), \(A\), nor \(\theta\) change, the magnetic flux remains constant. C. If the loop moves at constant velocity in a direction parallel to the axis of the loop while remaining in the uniform magnetic field, its area and orientation relative to the magnetic field do not change. Again, \(B\), \(A\), and \(\theta\) remain constant, so the magnetic flux does not change. D. If the loop is rotated through 180 degrees about an axis through its center and in the plane of the loop, the angle \(\theta\) between the magnetic field direction and the normal to the loop remains zero (or \(180^\circ\), which is effectively the same for \(\cos(\theta) \) calculations). Therefore, the flux remains unchanged despite the rotation. E. None of the above cases change the flux. The correct answer is E. None of the above cases cause a change in the magnetic flux through the loop.