Modern Physics
Photoelectric Effect
PYQ-2023-Dual-Nature-Of-Matter-Q1, PYQ-2023-Dual-Nature-Of-Matter-Q5, PYQ-2023-Dual-Nature-Of-Matter-Q11, PYQ-2023-Dual-Nature-Of-Matter-Q12
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work function: $$\mathrm{W}=h v_{0}=\frac{\mathrm{hc}}{\lambda_{0}}$$
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Work Function is Minimum For Cesium: $(1.9 \mathrm{eV})$
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Photoelectric current is directly proportional to intensity of incident radiation. $(v-$ constant $)$
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Photoelectrons ejected from metal have kinetic energies ranging from 0 to $\mathrm{KE}_{\max }$.
Here, $$KE_{max}=e V_s \quad V_s - \text{stopping potential}$$
PYQ-2023-Dual-Nature-Of-Matter-Q4
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Stopping potential Stopping potential is independent of intensity of light used ( $v$-constant)
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Intensity in the terms of electric field is $$ \mathrm{I}=\frac{1}{2} \in_{0} \mathrm{E}^{2} \cdot \mathrm{c} $$
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Momentum of one photon is $$p = \frac{h}{\lambda}$$
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Einstein equation for photoelectric effect is
PYQ-2023-Dual-Nature-Of-Matter-Q13
$$hv=w_0+k_{max}\Rightarrow \hspace{2mm}\frac{hc}{\lambda}=\frac{hc}{\lambda_0}+eV_s$$
- Energy:
PYQ-2023-Dual-Nature-Of-Matter-Q10, PYQ-2023-Atomic-Physics-Q4
$$\Delta \mathrm{E}=\frac{12400 \mathrm{eV}}{\lambda\left(\mathrm{A}^{0}\right)} $$
Angular momentum
PYQ-2023-Atomic-Physics-Q10, PYQ-2023-Motion-In-Two-Dimensions-Q4
$$ L = n \frac{h}{2\pi}$$
Force due to radiation (Photon) (no transmission)
PYQ-2023-Dual-Nature-Of-Matter-Q8
(i) When light is incident perpendicularly
(a) $$\quad a=1 \quad r=0$$
$$ \mathrm{F}=\frac{\mathrm{IA}}{\mathrm{C}}, \quad \text { Pressure }=\frac{\mathrm{I}}{\mathrm{C}} $$
(b) $$\quad r=1, \quad a=0$$
$$ F=\frac{2 I A}{c}, \quad P=\frac{2 I}{c} $$
(c) when $$0<r<1 \quad\text{and}\quad a+r=1$$
$$ F=\frac{I A}{c}(1+r), P=\frac{I}{c}(1+r) $$
When light is incident at an angle $\theta$ with vertical.
(a) $$ a=1,\quad r=0$$
$$ F=\frac{I A \cos \theta}{C}, \quad P=\frac{F \cos \theta}{A}=\frac{I}{C} \cos 2 \theta $$
(b) $$ r=1,\quad a=0$$
$$ F=\frac{2 I A \cos ^{2} \theta}{c}, \quad P=\frac{2 I \cos ^{2} \theta}{c} $$
(c) $$0<r<1, \quad a+r=1$$
$$ P=\frac{I \cos ^{2} \theta}{C}(1+r) $$
De Broglie wavelength:
PYQ-2023-Dual-Nature-Of-Matter-Q2, PYQ-2023-Dual-Nature-Of-Matter-Q3, PYQ-2023-Dual-Nature-Of-Matter-Q6, PYQ-2023-Dual-Nature-Of-Matter-Q7, PYQ-2023-Dual-Nature-Of-Matter-Q9
$$ \lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\mathrm{P}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mKE}}} $$
Radius And Speed Of Electron In Hydrogen Like Atoms:
PYQ-2023-Atomic-Physics-Q5, PYQ-2023-Atomic-Physics-Q7
$$r_n=\frac{n^2}{Z}a_0, \quad a_0=0.529 \stackrel{\circ}{A}$$
$$V_n=\frac{Z}{n}V_0, \quad V_0=2.19\times10^6 m/s$$
Energy In nth Orbit:
$$ E_{n}=E_{1} \cdot \frac{Z^{2}}{n^{2}} \quad E_{1}=-13.6 \mathrm{eV} $$
Wavelength Corresponding To Spectral Lines:
PYQ-2023-Atomic-Physics-Q2, PYQ-2023-Atomic-Physics-Q3, PYQ-2023-Atomic-Physics-Q6, PYQ-2023-Atomic-Physics-Q11
$$ \frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right] $$
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Lyman series: $\mathrm{n}_1=1,$ $\mathrm{n}_2=2,3,4$
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Balmer series: $\mathrm{n}_1=2,$ $\mathrm{n}_2=3,4,5$.
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Paschen series: $\mathrm{n}_1=3,$ $\mathrm{n}_2=4,5,6$
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The lyman series is an ultraviolet and Paschen, Brackett and Pfund series are in the infrared region.
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Total number of possible transitions, is $\frac{\mathrm{n}(\mathrm{n}-1)}{2}$, (from nth state)
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If effect of nucleus motion is considered,
$$ r_{n} = \left(0.529 , \text{\AA}\right) \frac{n^{2}}{Z} \cdot \frac{m}{\mu} $$
$$E_{n}=(-13.6 \mathrm{eV}) \frac{Z^{2}}{n^{2}} \cdot \frac{\mu}{m}$$
Here $\mu$ is reduced mass: $$\mu=\frac{M m}{(M+m)}, M-\text { mass of nucleus }$$
Minimum wavelength for $x$-rays:
$$\lambda_{min}=\frac{hc}{eV_0}=\frac{12400}{V_0(volt)}\stackrel{\circ}{A}$$
Moseley’s Law:
$$\sqrt{v}=a(z-b)$$
where: $a$ and $b$ are positive constants for one type of $x$-rays (independent of $z$ )
Average Radius Of Nucleus:
PYQ-2023-Nuclear-Physics-Q3, PYQ-2023-Nuclear-Physics-Q7
$$R=R_{0} A^{1 / 3}, \quad R_{0}=1.1 \times 10^{-15} M$$
$$A \text { - mass number }$$
Binding energy of nucleus of mass $M$:
PYQ-2023-Nuclear-Physics-Q2, PYQ-2023-Atomic-Physics-Q9, PYQ-2023-Atomic-Physics-Q12
$$B=\left(Z M_{p}+N M_{N}-M\right) C^{2}$$
Alpha - Decay Process:
PYQ-2023-Nuclear-Physics-Q1, PYQ-2023-Nuclear-Physics-Q5
$$^A_ZX\rightarrow\frac{A-4}{Z-2}Y+^4_2He$$
$$Q=\left[m(^A_ZX)-m\left(\frac{A-4}{z-2}Y\right)-m(^4_2He)\right]C^2$$
Beta- minus decay
$$\begin{gathered} { } _Z^A X \rightarrow{ } _{Z+1}^A Y+\beta^{-}+v^{-} \ \text {Q-value }=\left[m\left({ } _z^A X\right)-m\left({ } _{Z+1}^A Y\right)\right] c^2 \end{gathered}$$
Beta plus-decay
$$\begin{aligned} & { } _z^A X \longrightarrow{ } _{Z-1}^A Y+\beta+ v^{+} \ & Q-\text { value }=\left[m\left({ } _z^A X\right)-m\left({ } _{Z-1}^A Y\right)-2 m e\right] c^2 \end{aligned}$$
Electron capture : when atomic electron is captured, $x$-rays are emitted.
$$^A_zX+e\rightarrow\frac{A}{Z-1}Y+v$$
Radioactive Decay:
PYQ-2023-Nuclear-Physics-Q4, PYQ-2023-Nuclear-Physics-Q6, PYQ-2023-Nuclear-Physics-Q8, PYQ-2023-Nuclear-Physics-Q9
In radioactive decay, number of nuclei at instant $t$ is given by $$N=N_{0} e^{-\lambda t}$$
where: $\lambda$ is decay constant.
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Activity of sample : $$\quad A=A_{0} e^{-\lambda t}$$
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Activity per unit mass is called specific activity.
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Half life : $$T_{1 / 2}=\frac{0.693}{\lambda}$$
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Average life : $$T_{av}=\frac{T_{1/2}}{0.693}$$
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A radioactive nucleus can decay by two different processes having half lives $t_{1}$ and $t_{2}$ respectively.
Effective half-life of nucleus is given by $$\frac{1}{t}=\frac{1}{t_{1}}+\frac{1}{t_{2}}$$