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 is determined by the right-hand rule for the Lorentz force. The magnetic force \( \vec{F} \) on a charged particle is defined as: \[ \vec{F} = q \vec{V} \times \vec{B} \] where \( \vec{V} \) is the particle's velocity and \( \vec{B} \) is the magnetic field. This force is consistently perpendicular to both the velocity and the magnetic field.

At point P, the proton moves rightward (towards Q). For the magnetic force to be perpendicular to this velocity, if the deflection is upward, the magnetic field must be directed out of the paper (towards the observer).

At point Q, the proton's trajectory curvature, similar to P, implies the magnetic field is still directed out of the paper.

At point R, the proton moves downward. The resulting force on the proton indicates that the magnetic field is likely also directed out of the paper.

Concerning the magnetic field's magnitude, a more rapid change in the proton's velocity suggests a stronger magnetic field. Consequently, the magnetic field is inferred to be strongest at P, followed by Q, and weakest at R.

In summary, the magnetic field is directed out of the plane of the paper and exhibits its greatest strength at point P, decreasing at Q, and being weakest at R.