The Motion of Charged Particles in Magnetic Fields

Solutions

1) On your whiteboard, make four illustrations of the electrons' path, one for each of the following field strengths:
   (a) B = 0 inside the region marked by the dotted-line
   (b) weak B field inside the region marked by the dotted-line
   (c) medium strength B field inside the region marked by the dotted-line
   (d) very strong B field inside the region marked by the dotted-line

ANSWER:

An electron with velocity pointed toward the left side of the page is analogous to a positive charge moving in toward the right side of this page. Using RHR #1:

the electron feels a force pushing it down.

The four trajectories are as shown:

2) On your whiteboard, calculate the relative sizes of the two circular paths and sketch them to scale showing the direction of travel for each. Give explanations for your reasoning.

ANSWER:

Using RHR #1, the direction of the of the force due to the magnetic field is upwards:

A hydrogen ion is a proton so, q_H+= +e and m_H+ = mass of 1 proton.
An alpha particle is a helium nucleus so, q_He++ = +2e and m_He++ = 4m_H+.
Recall that F_centripetal= (mv²)/ r; where the direction of F is perpendicular to v, and
F_magnetic= qvB; where the direction of B is perpendicular to v. Using this, (mv²)/ r = qvB
(mv²)/ (qvB) = r = mv/ qB.

r_H+ = mv/ qB and r_He++ = 4mv/ 2qB; the radius of the He⁺⁺ circular trajectory is twice as large as the radius of the H⁺ circular trajectory.

Using this information:

where R_He++ = 2R_H+.

References:

Arons, A. (1996), Teaching Introductory Physics, Wiley: NY, p. 105.

Cutnell, J.and Johnson, K. (1998), Physics, Vol. 2, Wiley: NY, p. 634-6.

This page contributed to by Camilo Tafur and Dan MacIssac.

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