Thermionic emission – effect of a small temperature rise and Richardson–Dushman law Assertion (A): Raising the cathode temperature from 2500 K to 2550 K can increase thermionic emission current by roughly 50%. Reason (R): The emission current density obeys J ∝ T^2 * exp(−phi/(k T)).

Introduction / Context:
Thermionic emission is the process by which electrons escape from a heated metal or oxide cathode. The emitted current density depends sensitively on temperature through an exponential factor, which explains why even a small temperature rise can yield a large current increase in vacuum tubes and electron guns.

Given Data / Assumptions:

• Initial temperature T1 = 2500 K, final temperature T2 = 2550 K.
• Richardson–Dushman relation: J = A T^2 * exp(−phi/(k T)), where phi is work function and k is Boltzmann constant.
• Typical metallic work functions are around 4 to 5 eV.

Concept / Approach:

The T^2 prefactor and the strong exponential term with 1/T together control sensitivity. A small increase in T reduces the exponent magnitude and increases the T^2 term, producing a substantial rise in J. This quantitatively supports the assertion of a roughly 50% increase for a 50 K rise at around 2500 K.

Step-by-Step Solution:

Verification / Alternative check:

Changing phi between 4 and 5 eV still yields a change on the order of 40–70%, validating the claim's order of magnitude.

Why Other Options Are Wrong:

Since the numerical estimate aligns with the law, both A and R are true and R directly explains A; other combinations do not match the physics.

Common Pitfalls:

Ignoring the exponential sensitivity, or assuming linear behavior with temperature. Also, confusing cathode temperature limits with emission saturation due to space charge, which is a separate effect.

Final Answer:

Both A and R are true and R is the correct explanation of A