In a klystron amplifier, if the direct current (dc) electron velocity is v0 and the dc electron charge density is ρ0, then what is the expression for the dc beam current?

Introduction / Context:
Beam current in electron devices depends on the product of charge density and velocity. This relation is fundamental in electron beam theory for microwave tubes.

Given Data / Assumptions:

• Electron velocity = v0.
• Charge density = ρ0.

Concept / Approach:

Beam current density J = ρ * v. If beam cross-sectional area = 1 unit, then total current I = ρ0 * v0. This is the simplest form of the relation.

Step-by-Step Solution:

Verification / Alternative check:

All beam theory references confirm I = ρ0 * v0 * A, with A = 1 in this simplified form.

Why Other Options Are Wrong:

• ρ0 / v0, v0 / ρ0, (v0 / ρ0)^2: incorrect dimensional analysis.
• ρ0 * v0^2: introduces extra power of velocity.

Common Pitfalls:

Forgetting beam area factor; misinterpreting density definition.

Final Answer:

ρ0 * v0