Physics 132-1 Test 3

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Questions (8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. In the nuclear decay shown below what is the missing particle? Explain your reasoning.

   
   $\rm {^{210}Th \rightarrow {^4 He} } ~+~ ?$

   
   
   

2. The decay of atomic nuclei is often characterized by a quantity known as the half-life . The half-life is the period of time for one-half of the original sample to disappear via radioactive decay. This statement can be expressed mathematically in the following way

   
   

   $$N_{nuc}(t=\tau )=\frac{N_{0}}{2}$$
   
   

   where $N_{nuc}$ is the expression describing radioactive decay. Starting with the above expression show that the decay constant and the half-life are related by the following equation.

   
   

   $$\tau =\frac{\ln 2}{\lambda }$$
   
   

   
   
   
   

3. What is a wave?

   
   
   
   

4. How can an electromagnetic wave propagate across empty space?

   
   
   
   
   

5. Consider a laser beam shining on a pair of narrow slits. What would you see on a screen downstream from the slits if light were made of corpuscles?

   
   
   
   

6. The figure below shows the bright fringes that lie within the central diffraction envelope in two different double-slit diffraction experiments using the same wavelength of light. Is the slit size or width $a$ greater in experiment $A$ or $B$? Is the slit separation $d$ greater in experiment $A$ or $B$? Explain your reasoning.

   
   
   

   

   
   
   
   
   
   
   
   

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1.

15 pts.

A radioactive isotope of mercury, Hg, decays into gold, Au, with a disintegration constant of 0.0108 h . (a) What is its half-life? (b) What fraction of the original amount will remain after five half-lives? (c) What fraction will remain after 12.0 days?

2.	17 pts.	Measurements are made of the intensity distribution in a Young's, double-slit, interference pattern (see figure below). At a particular value of $y$, it is found that $I/I_{max} = 0.75$ when $\lambda = 650~nm$ light is used. What wavelength of light should be used so the relative intensity at the same location is reduced to $I/I_{max} = 0.65$?
3.	20 pts.	The Impressionist painter Georges Seurat created paintings like the one shown below with an enormous number of dots of pure pigment, each of which was approximately $r=1.0\times 10^{-3}~m$ in radius. The idea was to locate colors such as red and green next to each other to form a scintillating canvas. Outside what distance would one be unable to discern dots on the canvas? Assume that $\lambda = 5.0 \times 10^{-7}~m$ and the pupil diameter is $a = 4.0\times 10^{-3} m$.

Some useful constants, conversion factors, integrals, and other equations.

Coulomb's Law constant ( $k_e = {1 \over 4\pi\epsilon_0}$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Electron mass	$9.11\times 10^{-31}~kg$
Elementary charge ($e$)	$1.6\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Permittivity constant ($\epsilon_0$)	$8.85\times 10^{-12} {kg^2\over N-m^2}$	$\rm 1.0~eV$	$\rm 1.6\times 10^{-19}~J$
$1 ~ u$	$1.67\times 10^{-27}~kg$

$$\begin{eqnarray*} \int { dx \over (x^2 + a^2 )^{3/2}} = {x \over a^2(x^2 + a^2)^... ...[5pt] \int {x dx \over x+d} = x - d\ln (x+d) \qquad & & \\ [5pt] \end{eqnarray*}$$

$$\vec F = m \vec a \quad \vec F_B = q \vec v \times \vec B \q... ...quad \vec F_E = q\vec E \quad \vert\vec a\vert = {v^2 \over r}$$

$${d N \over d t} = - \lambda t \quad N = N_0 e^{-\lambda t} \quad t_{1/2} = \frac{\rm ln 2}{\lambda}$$

$$y = A \cos (kx - \omega t) \quad k\lambda = 2 \pi \quad \omega T = 2 \pi \quad \frac{\lambda}{T} = v_{wave}$$

$$E = E_m \sin (kx - \omega t) \quad B = B_m \sin (kx - \omega t) \quad I \propto {Amplitude}^2$$

$$I = I_m \cos^2 \left ( {\pi d \over \lambda} \sin \theta \ri... ...a \right )}{\frac{\pi a}{\lambda} \sin \theta}\right ]^2 \quad$$

$$I = I_m \cos^2 \left ( {\pi d \over \lambda} \sin \theta \ri... ... \theta \right )}{\frac{\pi a}{\lambda} \sin \theta}\right ]^2$$

$$\delta = d \sin \theta = m \lambda \quad \delta = a \sin \th... ...a \quad \phi = k\delta \quad \sin \theta_R = {\lambda \over a}$$

$${d \over dx}(x^n) = nx^{n-1} \qquad {d \over dx}(f(u)) = {df... ... du}{du \over dx} \qquad \int x^n dx = {x^{n+1} \over n+1} + c$$

$${d \over dx} (\cos ax) = -a\sin ax \qquad {d \over dx} (\sin ax) = a\cos ax$$

$$KE_0 + PE_0 = KE_1 + PE_1 \quad KE = {1 \over 2} m v^2 \quad PE = qV$$

The Periodic Chart