Question

A proton moving with velocity \( V \) 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:

Use the right-hand rule to determine the direction of the magnetic field based on the direction of motion of a positively charged particle.

Answer 1

Proton Path Analysis in a Non-uniform Magnetic Field

Provided Information:

• A proton (positive charge) travels with velocity \( \vec{V} \) within the plane of the paper.
• The proton's trajectory exhibits curvature, indicating the presence of a non-uniform magnetic field \( \vec{B} \).
• The objective is to ascertain the direction and relative strength of \( \vec{B} \) at specific points P, Q, and R.

Fundamental Principles:

• The magnetic force exerted on a moving charge is defined by: \[ \vec{F} = q (\vec{V} \times \vec{B}) \]
• For motion along a curved path, the centripetal force requirement can be met by the magnetic force. The radius of curvature \( r \) is related to the magnetic field strength \( B \) by: \[ \text{Centripetal force} = \frac{mv^2}{r} = qvB \Rightarrow r = \frac{mv}{qB} \] Consequently, a smaller radius of curvature implies a stronger magnetic field.
• The direction of the magnetic field can be deduced using the right-hand rule: orient the fingers of your right hand in the direction of \( \vec{V} \), curl them towards the direction of the magnetic force (indicated by the path's curvature), and your thumb will point in the direction of \( \vec{B} \).

Solution:

➡ Magnetic Field Direction:

The proton's path curves to the left. For a positively charged particle, this implies the magnetic force is directed towards the center of curvature. Applying the right-hand rule, using the velocity vectors at points P, Q, and R, we determine that:

• At all points P, Q, and R, the magnetic field \( \vec{B} \) is oriented perpendicular to the plane of motion and directed into the page (represented by the symbol “×”).

➡ Relative Magnetic Field Strength:

The relationship between the radius of curvature and the magnetic field strength is inverse, as shown by: \[ r = \frac{mv}{qB} \Rightarrow B \propto \frac{1}{r} \] Therefore, a smaller radius of curvature corresponds to a stronger magnetic field.

Observing the provided diagram:

• At point Q, the radius of curvature is the smallest, indicating that the magnetic field \( B_Q \) is the strongest.
• At point R, the radius of curvature is intermediate, implying a moderate magnetic field strength \( B_R \).
• At point P, the radius of curvature is the largest, signifying the weakest magnetic field strength \( B_P \).

✔ Conclusive Summary:

• Direction of \( \vec{B} \) at P, Q, R: Into the page
• Relative Magnitudes: \( B_Q > B_R > B_P \)