A circular coil with area A and N turns is free to rotate...

A circular coil with area $A$ and $N$ turns is free to rotate about a diameter that coincides with the $x$ -axis. Current $I$ is circulating in the coil. There is a uniform magnetic field $\vec { B }$ in the positive y-direction. Calculate the magnitude and direction of the torque $\vec { \tau }$ and the value of the potential energy $U ,$ as given in Eq. $( 27.27 ) ,$ when the coil is oriented as shown in parts (a) through (d) of Fig. E27.48.

A circular coil with area $A$ and $N$ turns is free to
rotate about a diameter that
coincides with the $x$ -axis.
Current $I$ is circulating in the
coil. There is a uniform magnetic field $\vec { B }$ in the positive y-direction. Calculate the magnitude and direction of the torque $\vec { \tau }$ and the value of the potential energy $U ,$ as given in Eq. $( 27.27 ) ,$
when the coil is oriented as shown in parts (a) through (d) of Fig.
E27.48.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment represents the strength and orientation of a current loop’s magnetic effect. For a coil with N turns carrying current I, it is given by the product of N, I, and the area vector A. The direction of the dipole moment is determined by the right?hand rule and is perpendicular to the plane of the coil, playing a central role in determining both the torque and the potential energy in a magnetic field.

Torque on a Magnetic Dipole

The torque experienced by a magnetic dipole in an external magnetic field is produced by the interaction of the magnetic dipole moment with the field. Mathematically, it is given by the cross product ? = m × B, which implies that the magnitude is ? = mB sin(?), where ? is the angle between the dipole moment and the magnetic field. This torque tends to align the dipole moment with the magnetic field direction.

Potential Energy of a Magnetic Dipole

The potential energy associated with a magnetic dipole in an external magnetic field is given by U = -m ? B. This scalar product indicates that the energy is minimized (most negative) when the dipole moment is aligned with the magnetic field. The concept is essential in understanding the work done by or against the torque when changing the orientation of the dipole in the field.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of a magnetic dipole, it is used to calculate the torque (? = m × B). The magnitude of the cross product gives the product of the magnitudes of the vectors and the sine of the angle between them, providing both a quantitative and directional determination of the torque.

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