Multicavity klystron amplifier: The order of magnitude of the electron beam velocity inside the device is approximately (choose the closest order).

Introduction:
Klystron amplifiers use a velocity-modulated electron beam that interacts with resonant cavities to achieve microwave gain. Estimating the order of the electron velocity helps in understanding transit time, bunching distance, and cavity spacing for practical designs.

Given Data / Assumptions:

• Typical beam voltages in classical klystrons are a few to tens of kilovolts.
• Operation is usually non-relativistic to mildly relativistic.
• We only need an order-of-magnitude estimate.

Concept / Approach:

The non-relativistic electron speed after acceleration through voltage Vb is v ≈ sqrt(2 * e * Vb / m). For Vb in the kV to tens of kV range, this speed is several million meters per second, i.e., around 10^6–10^7 m/s, well below the speed of light.

Step-by-Step Solution:

Verification / Alternative check:

Relativistic corrections become important above ~50–100 kV but velocities still remain a fraction of c, consistent with the chosen order. Design texts and datasheets cite transit times consistent with v on this order.

Why Other Options Are Wrong:

• 10 m/s or 10^3 m/s: far too small for accelerated electron beams.
• 10^12 m/s: non-physical (exceeds c).
• 3 × 10^8 m/s: essentially c; typical klystrons operate well below relativistic speeds.

Common Pitfalls:

Memorizing a single number; the actual value depends on Vb, but the correct order is in the millions of meters per second.

Final Answer:

10^6 m/s