An experimental velocity filter employs parallel metal plates 2.5 cm apart to create…
PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Magnetism and Electromagnetism · Magnetism and Electromagnetism
Problem
An experimental velocity filter employs parallel metal plates 2.5 cm apart to create a uniform, adjustable electric field. A uniform magnetic field fills the same space and is perpendicular to the plates. A beam of singly ionized potassium ions (mass 6.5×10⁻²⁶ kg) enters the device horizontally midway between the plates and emerges at the far end only if its speed lies within a narrow band centered on 9.0×10⁴ m s⁻¹.
(1) Derive an expression for the width Δv of the velocity band that can pass straight through when the electric field magnitude can be adjusted by at most ±3 % from its nominal value, while the magnetic field magnitude is held fixed.
(2) For B = 0.075 T and nominal ion speed 9.0×10⁴ m s⁻¹, compute the required nominal electric field magnitude and the numerical value of Δv.
(3) The ion source drifts slightly, producing ions with speeds uniformly distributed between 8.8×10⁴ m s⁻¹ and 9.2×10⁴ m s⁻¹. Calculate the fraction of ions that will successfully pass through the filter under the conditions of part (b).
(4) To enhance resolution, the engineer proposes doubling the plate separation while keeping the maximum allowable plate voltage unchanged. Analyze quantitatively how this modification affects (i) the nominal ion speed that is selected and (ii) the velocity bandwidth Δv. Provide clear reasoning for each conclusion.