A tungsten (Z = 74) target is bombarded by electrons in an x-ray tube. The K, L, and M energy levels for tungsten have the energies 69.5, 11.3, and 2.30 keV, respectively. (a) What is the minimum value of the accelerating potential that will permit the production of the characteristic

$$K_{\alpha}$$

and

$$K_{\beta}$$

lines of tungsten? (b) For this same accelerating potential, what is

$${\lambda}_{min}$$

? What are the (c)

$$K_{\alpha}$$

and (d)

$$K_{\beta}$$

wavelengths?

Verify that if $\hat{\alpha}_i, \hat{\beta}_j$, and $\hat{\gamma}_{i j}$ are as given by the mentioned equations, then $E(\hat{\mu})=\mu, E\left(\hat{\alpha}_i\right)=\alpha_i, E\left(\hat{\beta}_j\right)=$ $\beta_j$, and $E\left(\hat{\gamma}_{i j}\right)=\gamma_{i j}$ for all values of $i$ and $j$.

Beta Industries has net income of $\$ 2,000,000$, and it has $1,000,000$ shares of common stock outstanding. The company's stock currently trades at $\$ 32$ a share. Beta is considering a plan in which it will use available cash to repurchase $20 \%$ of its shares in the open market. The repurchase is expected to have no effect on net income or its stock price. What will be Beta's EPS following the stock repurchase?

A mutual fund manager has a $20 million portfolio with a beta of 1.7. The risk-free rate is 4.5%, and the market risk premium is 7%. The manager expects to receive an additional$5 million, which she plans to invest in a number of stocks. After investing the additional funds, she wants the fund’s required return to be 15%. What should be the average beta of the new stocks added to the portfolio?

Show that at high temperature, such that $k_{\mathrm{B}} T \gg \hbar \omega$, the partition function of the simple harmonic oscillator is approximately $Z \approx(\beta \hbar \omega)^{-1}$. Hence find U, C, F and S at high temperature. Repeat the problem for the high temperature limit of the rotational energy levels of the diatomic molecule for which $Z \approx\left(\beta \hbar^{2} / 2 I\right)^{-1}$.

A 65-kg person is accidentally exposed for 240 s to a 15-mCi source of beta radiation coming from a sample of $^{90}Sr.$ Each beta parti cle has an energy of $8.75 \times 10 ^ { - 14 } \mathrm { J }$ and 7.5% of the radiation is absorbed by the person. Assuming that the absorbed radiation is spread over the person's entire body, calculate the absorbed dose in rads and in grays.

Consider the differential equation $y^{\prime \prime}+\alpha y^{\prime}+\beta y=g(t)$. In each exercise, the complementary solution, $y_C(t)$, and nonhomogeneous term, $g(t)$, are given. Determine $\alpha$ and $\beta$ and then find the general solution of the differential equation.

$$y_C(t)=c_1 e^{-2 t}+c_2 t e^{-2 t}, \quad g(t)=5 \sin t$$

The values of the specific rotation, $[\alpha]_D^{20}$, for the $\alpha$ and $\beta$ anomers of D-galactose are $150.7^{\circ}$ and $52.8^{\circ}$, respectively. A mixture that is $20 \% \alpha$-D-galactose and $80 \% \beta$-D-galactose is dissolved in water at $20^{\circ} \mathrm{C}$. What is its initial specific rotation of this mixture reached an equilibrium value of $80.2^{\circ}$. What is its anomeric composition?

The differential equation $L[y]=t^{2} y^{\prime \prime}+\alpha t y^{\prime}+\beta y=0 \quad (*)$ is known as Euler's equation. Observe that $t^{2} y^{\prime \prime}, t y^{\prime},$ and y are all multiples of $t^r$ if $y=t^{r}.$ This suggests that we try $y=t^{r}$ as a solution of (). Show that $y=t^{r}$ is a solution of () if $r^{2}+(\alpha-1) r+\beta=0 .$

(a) Design a four-resistor bias network with the configuration to yield $Q$-point values of $I_{C Q}=50 \mu \mathrm{A}$ and $V_{C E Q}=5 \mathrm{~V}$. The bias voltages are $V^{+}=+5 \mathrm{~V}$ and $V^{-}=-5 \mathrm{~V}$. Assume a transistor with $\beta=80$ is available. The voltage across the emitter resistor should be approximately $1 \mathrm{~V}$. (b) The transistor in part (a) is replaced by one with $\beta=120$. Determine the resulting $Q$-point.

Cesium-137, a beta emitter, has a half-life of 30 y.

a. Write the balanced nuclear equation for the beta decay of cesium-137.

b. How many milligrams of a 16-mg sample of cesium- 137 would remain after 90 y ?

c. How many years are required for 28 mgof cesium- 137 to decay to 3.5 mg of cesium-137 ? hat is the age of the ancient wood object?

A bicycle wheel with a radius of $0.33 \mathrm{~m}$ turns with angular acceleration $\alpha(t)=\gamma-\beta t$, where $\gamma=1.80 \mathrm{rad} / \mathrm{s}^2$ and $\beta=$ $0.25 \mathrm{rad} / \mathrm{s}^3$. It is at rest at $t=0$, a) Determine the angular velocity and displacement as a function of time. b) Determine the wheel's maximum positive angular velocity and maximum positive angular displacement.

(a) The bias voltage in the circuit is changed to $V_{C C}=9 \mathrm{~V}$. The transistor current gain is $\beta=80$. Design the circuit such that $I_{C Q}=0.25 \mathrm{~mA}$ and $V_{C E Q}=4.5 \mathrm{~V}$. (b) If the transistor is replaced by a new one with $\beta=120$, find the new values of $I_{C Q}$ and $V_{C E Q}$. (c) Sketch the load line and $Q$-point for both parts (a) and (b).

Stock X has a $10 \%$ expected return, a beta coefficient of $0.9$, and a $35 \%$ standard deviation of expected returns. Stock Y has a $12.5 \%$ expected return, a beta coefficient of $1.2$, and a $25 \%$ standard deviation. The risk-free rate is $6 \%$, and the market risk premium is $5 \%$.

If the market risk premium increased to $6 \%$, which of the two stocks would have the larger increase in its required return?