An electron is moving in a uniform \(0.25-\) T magnetic field; its veIocity components parallel and perpendicular to the field are both 3.1 Mm/s. (a) What's the radius of the electron's spiral path? (b) How far does it move along the field direction in the time it takes to complete a full orbit about the field?

Short Answer

Expert verified
The radius of the electron's spiral path is approximately \(0.01\) m (or \(1\) cm), and the distance it moves along the field direction in the time it takes to complete a full orbit about the field is approximately \(19.63\) m.

Step by step solution

01

Calculate Radius of the Circular Path

In the presence of uniform magnetic field, an electron moving with a velocity component perpendicular to the field executes circular motion. The radius \(r\) of the circular path can be calculated using the expression for the Lorentz force, which equates to the centripetal force in this context. This gives the formula \(r = \frac{mv_{\perp}}{|q|B}\), with \(m\) being the mass of the electron, \(v_{\perp}\) the velocity component perpendicular to the field, \(|q|\) the absolute value of the electron's charge and \(B\) the magnetic field strength. Substituting the known values \(m = 9.11 \times 10^{-31}\) kg, \(v_{\perp} = 3.1 \times 10^6\) m/s, \(|q| = 1.6 \times 10^{-19}\) C, and \(B = 0.25\) T into the formula, we can calculate the radius.
02

Calculate the Distance Moved in Field Direction

The distance moved in the field direction, during one complete orbit, can be calculated using simple distance-velocity-time relation, i.e., \(d = v_{\parallel}T\). Here \(T = \frac{2\pi r}{v_{\perp}}\) is the time for one complete orbit (the period), \(v_{\parallel}\) is the electron's velocity component parallel to the field, and \(r\) is the radius calculated in step 1. Substituting the known values and the calculated radius, we find the distance.

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