The intrinsic magnetic dipole moment of the electron has...

The intrinsic magnetic dipole moment of the electron has magnitude $9.3 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2} .$ In other words, the electron acts as though it were a tiny current loop with $N I A=9.3 \times 10^{-24} \mathrm{A} \cdot \mathrm{m}^{2} .$ What is the maximum torque on an electron due to its intrinsic dipole moment in a $1.0-T$ magnetic field?

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Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the strength and orientation of a magnetic source, similar to how an electric dipole moment describes a pair of opposite charges. It can arise from current loops or intrinsic particle properties, and it fundamentally represents the ability of a system to interact with external magnetic fields.

Torque on a Magnetic Dipole

When a magnetic dipole is placed in an external magnetic field, it experiences a torque which tends to align the dipole with the field. Mathematically, this torque is given by the cross product of the dipole moment and the magnetic field vectors, and its magnitude is maximized when the dipole moment is perpendicular to the magnetic field, with the maximum torque being equal to the product of the magnetic dipole moment's magnitude and the magnetic field strength.

Vector Cross Product and Angular Dependence

The torque experienced by a magnetic dipole is evaluated using the vector cross product, which inherently includes a sine function of the angle between the dipole moment and the magnetic field. This means the torque depends on the angular orientation of the dipole relative to the field, reaching its maximum value when the angle is 90 degrees.

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