physics-class-12-unit-12-chapter-01-atomics-models-l-1-9-za8qzhtt4tk

### <Success style="font-size:25px"> Atomics Models L-1 </Success>
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- Source: ${ }_{83}^{214} \mathrm{Bi}$;

- Energy of $\alpha$ particles $5.5 \mathrm{MeV}$

- Target: A very thin gold foil $\left(2.1 \times 10^{-7} \mathrm{~m}\right)$

- Detector: Zinc Sulphide (scintillation) + Microscope

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<footer style = "font-size: 18px"> $\rightarrow$ Atomic Models $\rightarrow$ <span style = "color:blue"> Rutherford Experiment </span> $\rightarrow$ Modelling the Results $\rightarrow$ Rutherford Scattering </footer>

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- **Important features:**

- Most of the electrons are unscattered. They just pass through undeflected.
- The number of electrons getting back scattered is relatively quite large.

- **Basic assumption**

- Atom has a size of in the range of angstroms

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<footer style = "font-size: 18px">Atomic Models $\rightarrow$ Rutherford Experiment $\rightarrow$ <span style = "color:blue"> Modelling the Results </span> $\rightarrow$ Rutherford Scattering $\rightarrow$ Total Energy </footer>

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- **Au** 87 elentrons $\alpha$ particle

- $ \varepsilon=5.5 \mathrm{MeV}$

- $ Q_p$ + 87

- A = 199 $ \approx $ 200

- **Scattering from a fixed target.**

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<footer style = "font-size: 18px">Rutherford Experiment $\rightarrow$ Modelling the Results $\rightarrow$ <span style = "color:blue"> Rutherford Scattering </span> $\rightarrow$ Total Energy $\rightarrow$ Point Distribution </footer>

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- How close to the atom can an $\alpha$-particle get?
	
- **Almost point distribution**

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<footer style = "font-size: 18px">Rutherford Scattering $\rightarrow$ Total Energy $\rightarrow$ <span style = "color:blue"> Point Distribution </span> $\rightarrow$ Characterstics $\rightarrow$ Characterstics </footer>

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- **1.** Ignore the size of the Atom.

- **2.** Estimate $r_{\min }$

- **3.** Compare $r_{\text {min }}$ with $R$

- $\frac{Q_1 Q_2}{4 \pi \epsilon_0 r_{\min }}=5.5 \mathrm{MeV}$
- $ Q_1^\alpha=2 ; \quad Q_2^{A u}=87$

- $ r_{\text {min }}=\frac{Q_1 Q_2}{4 \pi \epsilon_0(5.5 \mathrm{MeV})}$
- $\frac{Q_1 Q_2}{4 \pi \epsilon_0 r_{\min }}=5.5 \mathrm{MeV}$
- $ Q_1^\alpha=2 ; \quad Q_2=87$

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<footer style = "font-size: 18px">Total Energy $\rightarrow$ Point Distribution $\rightarrow$ <span style = "color:blue"> Characterstics </span> $\rightarrow$ Characterstics $\rightarrow$ Potential </footer>

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- **If we plug in the numbers.**

- $ r_{min } \sim 10^{-14} m $

- $ R \sim 10^{-10} m.$

- $ \frac{r_{min }}{R} \sim 10^{-4}$

- $ V(r)=\frac{Q_1 Q_2}{4 \pi \epsilon_0 r} \quad \quad  r \geqslant R $

- $ \int \rho d^3 r=Q_2$

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<footer style = "font-size: 18px">Point Distribution $\rightarrow$ Characterstics $\rightarrow$ <span style = "color:blue"> Characterstics </span> $\rightarrow$ Potential $\rightarrow$ Potential </footer>

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- Inside the sphere, field rises linearly. Potential is quadratic in $r$.

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<img src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/babca674-0dea-40d4-a8aa-04f8f441b300/public">

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<footer style = "font-size: 18px">Characterstics $\rightarrow$ Characterstics $\rightarrow$ <span style = "color:blue"> Potential </span> $\rightarrow$ Potential $\rightarrow$ Potential of r </footer>

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- $V(r)=-\frac{1}{2} k r^2+\left.\left.c\right|_{r=R} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{\gamma}\right|_r$

- $k$ is fixed by equating forces at $r=R$.

- $k R=\frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R^2}$

- $ k  =\frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R^3}$

- $ V(r)  =-\frac{1}{8} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{r^2}{R^3}+{C}$

- $ if, r < R $

- $ =\frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R}  \quad \quad \text { if } r > R$

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<footer style = "font-size: 18px">Potential $\rightarrow$ Potential $\rightarrow$ <span style = "color:blue"> Potential of r </span> $\rightarrow$ Potential of r Outside The Sphere $\rightarrow$ Rutherford Scattering </footer>

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- $ -\frac{1}{8} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R}+C$ { Sphere( in side ) }

- $ =\frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R} $ (out side the sphere)

- $ \frac{1}{8}+\frac{1}{4}  =\frac{3}{8}$

- $ C  =\frac{3}{8} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R}$

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<footer style = "font-size: 18px">Potential $\rightarrow$ Potential of r $\rightarrow$ <span style = "color:blue"> Potential of r Outside The Sphere </span> $\rightarrow$ Rutherford Scattering $\rightarrow$ Rutherford Cross Section </footer>

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- $ V(r)=\frac{\frac{3}{8} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{R}-\frac{1}{8} \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{r^2}{R^3}}{r<R}$

- $r>R: \frac{Q_1 Q_2}{4 \pi \epsilon_0} \frac{1}{r}$

- $K=\frac{Q_1 Q_2}{4 \pi \epsilon_0} $

- **Essence of rutherford scattering**

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<footer style = "font-size: 18px">Potential of r $\rightarrow$ Potential of r Outside The Sphere $\rightarrow$ <span style = "color:blue"> Rutherford Scattering </span> $\rightarrow$ Rutherford Cross Section $\rightarrow$ Rutherford Experiment </footer>

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<footer style = "font-size: 18px">Rutherford Scattering $\rightarrow$ Rutherford Cross Section $\rightarrow$ <span style = "color:blue"> Rutherford Experiment </span> $\rightarrow$ Plum Pudding Model $\rightarrow$ Planetary Model </footer>

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### <primary style="font-size:24px"> Plum Pudding Model </primary>
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- The positive charge is distributed in a very small region in the atom

- $r_p \sim 10^{-4} r_{a t}$

- **Figure: Geiger marsden results**

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<footer style = "font-size: 18px">Rutherford Cross Section $\rightarrow$ Rutherford Experiment $\rightarrow$ <span style = "color:blue"> Plum Pudding Model </span> $\rightarrow$ Planetary Model $\rightarrow$ Electricity and Magnetism </footer>

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- $r<10^{-14} \mathrm{~m}$

- Planetary model

- $\frac{10^7 10^8 10^{10} \cdot m}{10^{-10}}$

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<footer style = "font-size: 18px">Rutherford Experiment $\rightarrow$ Plum Pudding Model $\rightarrow$ <span style = "color:blue"> Planetary Model </span> $\rightarrow$ Electricity and Magnetism $\rightarrow$ Synchrotron </footer>

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- **Electromagnetism**

- Electrodynamics is much more complicated than newtonian gravity.

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<footer style = "font-size: 18px">Plum Pudding Model $\rightarrow$ Planetary Model $\rightarrow$ <span style = "color:blue"> Electricity and Magnetism </span> $\rightarrow$ Synchrotron $\rightarrow$ Thank You </footer>

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<footer style = "font-size: 18px">Planetary Model $\rightarrow$ Electricity and Magnetism $\rightarrow$ <span style = "color:blue"> Synchrotron </span> $\rightarrow$ Thank You $\rightarrow$  </footer>