Problem 1

A particle of mass

m

and charge

q

moves perpendicular to a uniform magnetic field

B

at speed

v

. Find orbit radius and period; use

|q|

for charge magnitude. The magnetic field is perpendicular to the initial velocity; neglect gravity and collisions. (a) Use the Lorentz force to find orbit radius

r

. (b) Find the orbital period

T

. (c) Find the cyclotron frequency and identify which result is independent of speed

v

. Express each answer in terms of the supplied parameters, using SI units for any numerical evaluation. State the assumptions used, check dimensions at each result, and compare with the governing law to catch a sign error or a missing condition.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{r=\frac{mv}{|q|B},\quad T=\frac{2\pi m}{|q|B},\quad f=\frac{|q|B}{2\pi m}}

problems.proof.analysis

The final expressions are collected in the order requested and must satisfy the equations used above. Checking dimensions, signs, and limiting conditions catches algebra errors or omitted constraints.