The charge-to-mass ratio (q/m) of the particle can be determined by equating the magnetic force to the centripetal force and solving for q/m, using the velocity that is derived from the period of the circular path. The velocity of the particle is calculated using the circumference and the period. Lastly, the angle between the velocity and the magnetic field in a helical path can be found using trigonometric relations between the pitch of the helical motion and the circumference of the circular path.
To find the charge-to-mass ratio (q/m) of the particle, we use the relationship between the magnetic force and the centripetal force that keeps the particle moving in a circle:
- Fmagnetic = qvB
- Fcentripetal = mv2/r
Since these forces are equal for a particle moving in a circular path within a magnetic field, we can equate them:
qvB = mv2/r
By rearranging the formula, we can find the charge-to-mass ratio:
q/m = v/rB
Next, we find the velocity (v) because it's needed to calculate q/m. The velocity can be found with the following relationship, where T is the period of rotation:
v = 2πr/T
Plugging in values for the radius (r = 15.0 cm = 0.15 m) and the period (T = 2.5 ms = 2.5 x 10-3 s), we get:
v = (2π x 0.15 m) / (2.5 x 10-3 s)
Now with this velocity and the magnetic field strength (B = 3.0 T), we can calculate the charge-to-mass ratio q/m.
For the last part, to find the angle θ between the particle's velocity and the magnetic field in a helical path, we consider that the pitch of the helix is the distance the particle moves along the magnetic field direction in one cycle. Since we know the pitch (10.0 cm) and the circumference of the circle (2πr), we can use trigonometry to find the angle.
tan(θ) = pitch / circumference
θ = arctan(pitch / (2πr))
Calculating these values will give the results for (b) and (c).