The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function 4 eV. [2023]

Question

Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option. [2023]

[Given: Wien's constant as $2.9 \times 10^{- 3} m K$ and $\frac{h c}{e} = 1.24 \times 10^{- 6} V m$ ]

	List-I		List-II
(P)	2000 K	(1)	The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function 4 eV.
(Q)	3000 K	(2)	The radiation at peak wavelength is visible to human eye.
(R)	5000 K	(3)	The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction.
(S)	10000 K	(4)	The power emitted per unit area is 1/16 of that emitted by a blackbody at temperature 6000 K.
		(5)	The radiation at peak emission wavelength can be used to image human bones.

Smart Achievers · Accepted Answer

(3)

From Wien's displacement law, $λ_{m} T = b (constant)$

and $b = 2.9 \times 10^{- 3} mK$

When $T = 2000 K$ (P), $λ_{m} = \frac{b}{T} = \frac{2.9 \times 10^{- 3}}{2000} = 1450 nm (\max)$

When $T = 3000 K$ (Q), $λ_{m} = \frac{2.9 \times 10^{- 3}}{3000} = 966.66 nm$

When $T = 5000 K$ (R), $λ_{m} = \frac{2.9 \times 10^{- 3}}{5000} = 580 nm$

When $T = 10000 K$ (S), $λ_{m} = \frac{2.9 \times 10^{- 3}}{10000} = 290 nm (\min)$

For option (P), $λ$ maximum and $β = \frac{λ D}{d}$

$∴$ widest central maximum

For option (Q), Power

$P_{3000} = σ A {(3000)}^{4}$

$P_{6000} = σ A {(6000)}^{4}$

$\frac{P_{3000}}{P_{6000}} = {(\frac{1}{2})}^{4} = \frac{1}{16}$

$∴$ $P_{3000} = \frac{1}{16} P_{6000}$

For option (R), Wavelength $λ = 580 nm$

Visible to human eyes.

For (S), $λ = \frac{h c}{ϕ} = \frac{1.24 \times 10^{- 6}}{4} = 310 nm$

$290 nm < 310 nm$ so this radiation leads to emission of photoelectrons from a metal of work function $ϕ = 4 eV .$

For imaging human bones, X-rays of wavelength range 1 - 10 nm are used.

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