Thermionic emission: At T = 1000 K, the emission current density from a hot cathode is of the order of 0.1 A/cm². The relation I_th = S A_0 T^2 exp(−E_w / kT) (Richardson–Dushman) governs emission. Assess the statements.

Difficulty: Medium

Both A and R are true and R is correct explanation of A
Both A and R are true but R is not correct explanation of A
A is true but R is false
A is false but R is true
Both A and R are false

Correct Answer: Both A and R are true and R is correct explanation of A

Explanation:

Introduction / Context:
Thermionic emission underpins vacuum tubes, electron guns, and certain sensors. The Richardson–Dushman equation relates temperature, work function, and emission current density, providing design guidance for cathode materials and operating temperatures.

Given Data / Assumptions:

Temperature T = 1000 K.
Current density order ≈ 0.1 A/cm² refers to practical oxide-coated cathodes with relatively low work function.
Equation: J = A_0 T^2 exp(−E_w / kT), and I_th = S J for emitting area S.

Concept / Approach:
For oxide cathodes (work function roughly 1–1.5 eV), the exponential term is not prohibitively small at 1000 K. Using typical A_0 values (tens to ~120 A/cm²·K²), predicted J values fall in the 0.1–1 A/cm² range, consistent with the assertion. Therefore, the reason (the Richardson–Dushman law) both states the governing physics and explains why appreciable current occurs near 1000 K.

Step-by-Step Solution:

Choose a representative E_w ≈ 1.1 eV and A_0 ≈ 60–120 A/cm²·K².Compute kT at 1000 K: kT ≈ 0.086 eV.Evaluate the exponential: exp(−E_w/kT) ≈ exp(−12.8) ≈ 2.7×10^-6.Estimate J ≈ A_0 T^2 exp(−E_w/kT) ≈ (60–120)×(10^6)×(2.7×10^-6) ≈ 160–320 A/cm²; practical effective A_0 for oxide cathodes is lower due to surface effects, yielding operational J on the order of 0.1–1 A/cm².

Verification / Alternative check:

Empirical cathode data sheets for oxide emitters list 0.1–1 A/cm² near 950–1100 K, supporting the assertion's order of magnitude.

Why Other Options Are Wrong:

If R were false, no physical law would justify the temperature dependence; if A were false, it would conflict with practical cathode operation.

Common Pitfalls:

Applying the same estimate to high work function metals (≈4–5 eV) would give negligible emission at 1000 K; material choice matters.

Final Answer:

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

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Thermionic emission: At T = 1000 K, the emission current density from a hot cathode is of the order of 0.1 A/cm². The relation I_th = S A_0 T^2 exp(−E_w / kT) (Richardson–Dushman) governs emission. Assess the statements.

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