From the surface of a cylindrical wire of radius R=1 cm carrying a current I=1 mA an electron escapes with velocity v0=30 m/s perpendicular to the surface of the wire. What will be the electron's maximum distance in meters from the axis of the wire?
Details and assumptions
μ04π=10−7 H/m
e/me=1.76×1011 C/kg
To determine the electron's maximum distance from the axis of the wire, we can use the concepts of magnetic forces acting on moving charges. The force experienced by a charged particle moving through a magnetic field is given by the equation F = qvB, where q is the charge of the particle, v is its velocity, and B is the magnetic field strength.
In this case, the electron escapes from the cylindrical wire surface, so it will experience a force due to the current flowing through the wire. The magnetic field at a distance r from the axis of a long straight wire carrying current I can be calculated using Ampere's Law as B = μ0I / (2πr), where μ0 is the magnetic constant.
Given that the radius of the wire (R) is 1 cm and the current (I) is 1 mA, we can substitute these values into the formula to find the magnetic field at the surface.
B = μ0I / (2πR)
= (10^-7 H/m) × (1 × 10^-3 A) / (2π × 0.01 m)
= (1 × 10^-10 T)
Now, to find the maximum distance (d) of the electron from the axis of the wire, we can use the equation F = qvB and apply it to the electron's motion perpendicular to the wire surface. The force acting on the electron will be the centripetal force keeping it in circular motion, which is given by F = (m × v^2) / r, where m is the mass of the electron.
(qvB) = (m × v^2) / r
(qvB) = (me × v^2) / r (substituting the mass of the electron, me)
qvB = (me × v^2) / r
Since the velocity (v) and the charge of the electron (q) are perpendicular to the magnetic field (B), the equation can be simplified as:
qvB = (me × v^2) / r
Rearranging the equation to solve for the maximum distance (d) from the axis, we get:
d = (me × v) / (q × B)
Substituting the given values:
d = (1.76 × 10^11 C/kg × 30 m/s) / (1.6 × 10^-19 C × 1 × 10^-10 T)
= (1.76 × 10^11 kg m/s) / (1.6 × 10^-29 kg m/C)
= 110 × 10^18 m
= 1.1 × 10^19 m
Therefore, the electron's maximum distance from the axis of the wire is approximately 1.1 × 10^19 meters.