A charge q is distributed uniformly around a thin ring of...

A charge $q$ is distributed uniformly around a thin ring of radius $r$. The ring is rotating about an axis through its center and perpendicular to its plane, at an angular speed $\omega$. (a) Show that the magnetic moment due to the rotating charge has magnitude $\mu=\frac{1}{2} q \omega r^{2} .$ (b) What is the direction of this magnetic moment if the charge is positive?

Key Concepts

Right-Hand Rule for Magnetic Moment Direction

The direction of the magnetic dipole moment is determined by the right-hand rule, which relates the circulating current or angular momentum vector to the orientation of the magnetic moment. For a positive charge rotating in a circle, pointing the thumb of the right hand in the direction of the angular velocity results in the fingers curling in the direction of the current, thereby setting the magnetic moment's direction along the thumb's direction.

Uniform Charge Distribution and Rotation

Uniform charge distribution on a ring means that the total charge is spread evenly along its circumference. When the ring rotates, every segment of charge contributes equally to creating a current around the loop. The symmetry of the problem simplifies calculations and ensures that the rotating charge behaves like a well-defined current loop with a magnetic moment determined by the charge, rotation frequency, and the ring's geometry.

Current Loop Equivalence

A rotating charged ring can be equivalently treated as a current loop because the movement of charge around the ring defines a circulating current. The current is calculated by considering the total charge passing a point in one complete rotation, with the magnitude being proportional to the rotating charge's angular speed. This equivalence allows one to use the standard formulas for the magnetic moment of a current loop.

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of a magnetic source. In systems like a current loop or a rotating charged body, it quantifies the system's ability to produce a magnetic field, and it is defined in terms of the current and geometry of the system. The magnetic dipole moment directly relates to observable phenomena such as torque in external magnetic fields.

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