Dual Nature of Matter

JEE Main level, single correct, medium

147 of 200

Question 147 of 200

Let K1{K}_{1} and K2{K}_{2} be the maximum kinetic energies of photo-electrons emitted when two monochromatic beams of wavelength λ1{\lambda }_{1} and λ2{\lambda }_{2}, respectively are incident on a metallic surface. If λ1=3λ2{\lambda }_{1}=3{\lambda }_{2} then :

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Answer: (B) K1<K23{K}_{1}\lt \dfrac{{K}_{2}}{3}

According to Einstein's photoelectric equation

h⁡cλ=ϕ0+KEmax\dfrac{h⁡c}{\lambda }={\phi }_{0}+{\text{KE}}_{\text{max}}, where ϕ0{\phi }_{0} is a work function of a metal

∴K⁡1=h⁡cλ1−ϕ0…(i)∴ {K⁡}_{1}=\dfrac{h⁡c}{{\lambda }_{1}}-{\phi }_{0} \ldots \left(\text{i}\right)

K⁡2=h⁡cλ2−ϕ0…(ii){K⁡}_{2}=\dfrac{h⁡c}{{\lambda }_{2}}-{\phi }_{0} \ldots \left(\text{i}\text{i}\right)

orK1−K2=hc[1λ1−1λ2]\text{or} {K}_{1}-{K}_{2}=hc\left[\dfrac{1}{{\lambda }_{1}}-\dfrac{1}{{\lambda }_{2}}\right]

=hc[13λ2−1λ2]=−2hc3λ2(Givenλ1=3λ2)=hc\left[\dfrac{1}{{3\lambda }_{2}}-\dfrac{1}{{\lambda }_{2}}\right]=-\dfrac{2hc}{3{\lambda }_{2}} \left(\text{Given} {\lambda }_{1}=3{\lambda }_{2}\right)

=−23(K⁡2+ϕ0)(Using (ii))=-\dfrac{2}{3}\left({K⁡}_{2}+{\phi }_{0}\right) \left(\text{Using (ii)}\right)

or⁡K⁡1=K⁡2−23K⁡2−23ϕ0=K⁡23−23ϕ0or⁡ {K⁡}_{1}={K⁡}_{2}-\dfrac{2}{3}{K⁡}_{2}-\dfrac{2}{3}{\phi }_{0}=\dfrac{{K⁡}_{2}}{3}-\dfrac{2}{3}{\phi }_{0}

orK⁡1<K⁡23\text{or} {K⁡}_{1}\lt \dfrac{{K⁡}_{2}}{3}

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