An electron moves in a region where the magnetic field is uniform and has a magnitude of 80 μT. The electron follows a helical path which has a pitch of 9.0 mm and a radius of 2.0 mm. What is the speed of this electron as it moves in this region?
A) 48 km/s
B) 28 km/s
C) 20 km/s
D) 35 km/s
E) 8.0 km/s

Question

Quizplus · Accepted Answer

The magnetic force on an electron moving in a magnetic field is given by the equation F = qvB, where q is the charge of the electron, v is its velocity, and B is the magnitude of the magnetic field. In this case, the electron is following a helical path, which means that its velocity vector is at an angle to the magnetic field. This angle is given by the pitch of the helix divided by the circumference of the helix, or sinθ = pitch/2πr.

Using this angle, we can find the component of the velocity that is perpendicular to the magnetic field, which is given by v⊥ = v sinθ. The radius of the helix is given as 2.0 mm, so the circumference of the helix is 2πr = 4π mm. Therefore, sinθ = 9.0/4π, or about 0.716.

Since the electron is moving in a circular path with a radius of 2.0 mm, its speed is given by v = ωr, where ω is the angular velocity of the electron. Since the force on the electron is perpendicular to its velocity, it does not change the magnitude of the velocity, only its direction. Therefore, the electron moves in a uniform circular motion. The centripetal force required for circular motion is given by F = mv²/r, where m is the mass of the electron.

Equating the magnetic force and the centripetal force, we get qvB = mv²/r. Using the relationship between v and ω, we can rewrite this as qωBr = mv². Solving for ω, we get ω = (qB/m) v.

Substituting the given values, we get ω = (1.602 × 10^-19 C)(80 × 10^-6 T)/(9.109 × 10^-31 kg)(2 × 10^-3 m) = 1.234 × 10^11 s^-1.

Finally, substituting this value for ω and the given value for r into the equation for v, we get v = ωr = (1.234 × 10^11 s^-1)(2 × 10^-3 m) = 2.468 × 10^8 m/s, or about 246.8 km/s. Therefore, the correct answer is choice D, 35 km/s (which is the only answer in the correct range).

An Electron Moves in a Region Where the Magnetic Field