Figure 28-28 gives the orientation energy U of a magnetic...

Figure 28-28 gives the orientation energy $U$ of a magnetic dipole in an external magnetic field $\vec{B}$, as a function of angle $\phi$ between the directions of $\vec{B}$ and the dipole moment. The vertical axis scale is set by $U_{s}=2.0 \times 10^{-4} \mathrm{~J}$. The dipole can be rotated about an axle with negligible friction in order to change $\phi$. Counterclockwise rotation from $\phi=0$ yields positive vallues of $\phi$, and clockwise rotations yield negative values. The dipole is to be released at angle $\phi=0$ with a rotational kinetic energy of $9.0 \times 10^{-4} \mathrm{~J}$, so that it rotates counterclockwise. To what maximum value of $\phi$ will it rotate? (In the language of Module $8-3$, what value $\phi$ is the turning point in the potential well of Fig. $28-28 ?$ )

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Key Concepts

Magnetic Dipole Potential Energy

This concept describes the energy associated with a magnetic dipole in an external magnetic field, typically expressed as a function of the angle between the dipole moment and the field. The potential energy is minimum when the dipole is aligned with the field and increases as the dipole rotates away from alignment, defining a potential energy landscape that governs the dipole’s behavior.

Energy Conservation in Rotational Dynamics

In systems undergoing rotational motion, the principle of energy conservation states that the total energy (sum of kinetic and potential energies) remains constant in the absence of non-conservative forces. As the dipole rotates, its rotational kinetic energy is converted into potential energy until the kinetic energy is completely depleted at the turning point.

Turning Points in Potential Energy

A turning point is reached when a system’s kinetic energy drops to zero while it stores all its energy as potential energy. At this point, the direction of motion reverses. In the context of a rotating magnetic dipole, the turning point represents the maximum angle reached where further rotation would require additional energy input.

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