Effect of doubling the strength of the magnetic field on a bar magnet

a) When the strength of the external magnetic field is doubled while keeping it uniform, the following effects will be observed:

1. Increase in the torque: The torque experienced by a bar magnet in a uniform magnetic field is given by the equation τ = m'Bsinθ, where τ is the torque, m' is the magnetic moment of the bar magnet, B is the magnetic field strength, and θ is the angle between the magnetic moment and the magnetic field. As the magnetic field strength (B) is doubled, the torque acting on the bar magnet will also double.

2. Increase in the angular acceleration: The torque acting on a bar magnet causes it to rotate. According to Newton's second law of rotational motion, the torque is equal to the moment of inertia (I) multiplied by the angular acceleration (α). As the torque is doubled, the angular acceleration will also double.

3. Decrease in the time period of oscillation: The time period of oscillation for a bar magnet in a uniform magnetic field is given by the equation T = 2π√(I/τ), where T is the time period, I is the moment of inertia, and τ is the torque. As the torque is doubled, the time period of oscillation will decrease.

Change in potential energy of the magnet when rotating the magnetic dipole moment

b) When an external torque is applied to rotate the bar magnet from a position where the magnetic dipole moment is parallel to the magnetic field to a position where it is perpendicular to the field, the potential energy of the magnet will change. The change in potential energy can be explained as follows:

1. Initial position: In the initial position, where the magnetic dipole moment is parallel to the magnetic field, the potential energy of the magnet is minimum. This is because the angle between the magnetic dipole moment and the magnetic field is zero, and the potential energy is given by the equation U = -m'Bcosθ, where U is the potential energy, m' is the magnetic moment, B is the magnetic field strength, and θ is the angle between the magnetic moment and the magnetic field. As cosθ is equal to 1 when θ is zero, the potential energy is minimum.

2. Final position: In the final position, where the magnetic dipole moment is perpendicular to the magnetic field, the potential energy of the magnet is maximum. This is because the angle between the magnetic dipole moment and the magnetic field is 90 degrees, and the potential energy is given by the equation U = -m'Bcosθ. As cosθ is equal to 0 when θ is 90 degrees, the potential energy is maximum.

3. Change in potential energy: The change in potential energy (ΔU) can be calculated by subtracting the initial potential energy (U_initial) from the final potential energy (U_final), ΔU = U_final - U_initial. As the potential energy in the initial position is minimum and in the final position is maximum, the change in potential energy will be positive.

In conclusion, when rotating the bar magnet from a position where the magnetic dipole moment is parallel to the magnetic field to a position where it is perpendicular to the field, there will be an increase in potential energy of the magnet.

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