Question

A small charged ball suspended on an inextensible thread of length l moves in a uniform time-independent upward magnetic field of induction B. The mass of the ball is m, the charge is q, and the period of revolution is T.

Determine the radius r of the circle in which the ball moves if the thread is always stretched.

Answer 1

Let us choose the coordinate system xOy with the origin coinciding with the instantaneous position of the ball (Fig. 222).

The x-axis is "centripetal", while the y-axis is vertical just as the magnetic induction B.

The system of equations describing the motion of the ball (we assume that the ball moves in a circle counterclockwise) will be written in the form

Solving this system of equations, we obtain

The plus sign corresponds to the counterclockwise rotation of the ball, and the minus sign to the clockwise rotation (if we view from the top).