Drift velocity in uniform magnetic field near charged wire

[Wiseman101]

Problem
(a) Derive and expression (using cylindrical coordinates) for the drift velocity of a particle of mass m and charge q in a uniform vertical magnetic field B$\hat{z}$ when the particle is located at a perpindicular distance p from a long, straight vertical wire carrying a line charge density of λ.

(b) Obtain an expression for the radius p at which the drift velocity vanishes provided q and λ are of opposite sign, and verify that for λ = 10^-3, electrons experience zero net drift at p = 1cm when B = 1T.

Homework Equations

Drift velocity v = $\frac{F×B}{qB^2}$
The electic field due to a vertical wire carrying a charge density of λ is
$\bar{E}$ = ($λ/2\piερ$)$\hat{ρ}$

The Attempt at a Solution

(a) Using the formula with F = qE i obtained the drift velocity to be
v = ($-λ/2\piεB$)$\hat{θ}$

(b) So the electron experiences is rotating in a circle and experiences a radial force F = mρω² (where ω = drift velocity).
Then would you plug this back into the drift velocity equation to get a second drift term?
I tried this and set the total drift equal to zero. This didn't work out for the values of the charge density and radius given. Is my method correct so far? Thanks.

 

[mark726]

Solution:

(a) The drift velocity of a particle in a magnetic field can be expressed as v = (F x B)/qB^2, where F is the force on the particle, B is the magnetic field, and q is the charge of the particle. In this case, the particle is located at a perpendicular distance p from a long, straight vertical wire carrying a line charge density of λ. Using cylindrical coordinates, the force on the particle can be expressed as F = qE = q(λ/2περ)ρ̂, where ρ̂ is the unit vector in the radial direction. Therefore, the drift velocity can be written as:

v = (q(λ/2περ)ρ̂ x B)/qB^2 = (-λ/2πεB)ρ̂ x ρ̂ = (-λ/2πεB)θ̂

(b) To find the radius p at which the drift velocity vanishes, we can set the drift velocity equal to zero and solve for p:

(-λ/2πεB)θ̂ = 0
λ = 0

Since λ cannot be equal to zero, this means that the drift velocity will never vanish for any value of p. This is because the force on the particle, due to the electric field of the wire, will always be present and will always cause the particle to experience a drift velocity in the θ direction.

To verify that for λ = 10^-3, electrons experience zero net drift at p = 1cm when B = 1T, we can plug in these values into the drift velocity equation:

v = (-λ/2πεB)θ̂ = (-10^-3/2πε(1T))θ̂ = (-10^-3/2π(8.85x10^-12 C^2/Nm^2)(1T))θ̂ = (-1.13x10^-8 m/s)θ̂

At p = 1cm = 0.01m, the magnitude of the drift velocity is:

|v| = 1.13x10^-10 m/s

Since this value is very small, we can say that for λ = 10^-3 and B = 1T, the electrons will experience zero net drift at p = 1cm.