Question

A proton moving with velocity \( \vec{V} \) in a non-uniform magnetic field traces a path as shown in the figure. The path followed by the proton is always in the plane of the paper. What is the direction of the magnetic field in the region near points P, Q, and R? What can you say about relative magnitude of magnetic fields at these points?

Hint:

The right-hand rule helps determine the direction of the magnetic field when a charged particle moves in a magnetic field. The magnetic field is perpendicular to both the velocity and the force.

Answer 1

The direction of the magnetic field can be determined using the right-hand rule for the Lorentz force. The magnetic force \( \vec{F} \) on a charged particle is given by: \[ \vec{F} = q \vec{V} \times \vec{B} \] where \( \vec{V} \) is the velocity of the proton and \( \vec{B} \) is the magnetic field. The force is always perpendicular to the velocity and the magnetic field. 

At point P, since the proton is moving to the right (towards point Q), the magnetic force must act in a direction perpendicular to the velocity. If we assume that the proton is deflected upward at point P, the magnetic field at P must be directed out of the plane of the paper (towards the observer).

At point Q, the proton's path is curved in such a way that the magnetic field is still out of the plane, as indicated by the direction of deflection.
At point R, where the proton is moving downward, the force acting on the proton would indicate that the magnetic field is likely still out of the plane of the paper.

Regarding the magnitude of the magnetic field at these points, we can infer that the magnetic field is stronger where the proton's velocity changes more rapidly. Therefore, the magnetic field is likely strongest at point P, followed by point Q, and weakest at point R.
Thus, the magnetic field is directed out of the plane of the paper and is strongest at point P, followed by Q, and weakest at point R.