MikaStars39/nano-eval huggingface dataset:
Calculate the following products:
$$
\int_{|z|=1} \frac{\mathrm{d}z}{(z-a)^n (z-b)^n} \quad \left(\begin{array}{l} {|a| < 1} \\ {|b| < 1} \end{array}, a \neq b, n \text{ is a positive integer}\right)
$$

Options:
A. $$1$$
B. $$\frac{\pi}{(b-a)^n}$$
C. $$\frac{1}{(a+b)^{n+1}}$$
D. $$2$$
E. $$-1$$
F. $$\frac{2\pi i}{(a+b)^n}$$
G. $$\frac{(n-1)!}{(a-b)^n}$$
H. $$\frac{n!}{(a-b)^n}$$
I. $$\frac{2\pi a}{(b-a)^{2n}}$$
J. $$0$$
Answer: J
Find the value of the integral $\int_{0}^{\pi} \operatorname{tan}(\theta + \mathrm{i} a) \, \mathrm{d}\theta \quad (a$ is a real number and $a \neq 0)$.

Options:
A. $$
\pi i
$$
B. $$
2 \pi i
$$
C. $$
0
$$
D. $$
- \pi i
$$
E. $$
- 2 \pi i
$$
F. $$
i
$$
G. $$
\pi
$$
H. $$
- \pi
$$
I. $$
- i
$$
J. $$
2 i
$$
Answer: D
Using the residue theorem, calculate the integral
$$
\oint_{|z|=2} \frac{2z-1}{z(z-1)^2} \mathrm{d}z
$$

Options:
A. $$\pi i$$
B. $$2\pi i$$
C. $$0$$
D. $$-1$$
E. $$-4\pi i$$
F. $$-2$$
G. $$1$$
H. $$\frac{-\pi i}{2}$$
I. $$-2\pi i$$
J. $$\frac{1}{2}$$
Answer: C
Three pairs of corresponding points are provided with specific values: $z_{1} \leftrightarrow w_{1}$, $z_{2} \leftrightarrow w_{2}$, $z_{3} \leftrightarrow w_{3}$. Write out the corresponding fractional linear transformation, and specify what region the transformation maps the interior of the circle passing through $z_{1}, z_{2}, z_{3}$ or the left side of the line (as observed along $z_{1}, z_{2}, z_{3}$) into.
$$
1 \leftrightarrow 1, \mathrm{i} \leftrightarrow 0, -\mathrm{i} \leftrightarrow -1
$$

Options:
A. $$
{\frac{(1+2\mathrm{i})(z+1)}{1+z+2\mathrm{i}(1-z)}}
$$
B. $$
{\frac{(1+\mathrm{i})(z+1-\mathrm{i})}{2+z+\mathrm{i}(1-z)}}
$$
C. $$
{\frac{(1-\mathrm{i})(z-\mathrm{i})}{1-z+3\mathrm{i}(1+z)}}
$$
D. $$
{\frac{( 1+\mathrm{i} ) ( z-\mathrm{i} )} {1+z+3 \mathrm{i} ( 1-z )}}
$$
E. $$
{\frac{(1+\mathrm{i})(z+1)}{1+z+3(1-z+\mathrm{i})}}
$$
F. $$
{\frac{(1-\mathrm{i})(z-\mathrm{i})}{1-z-3\mathrm{i}(1+z)}}
$$
G. $$
{\frac{(1-\mathrm{i})(z+\mathrm{i})}{1-z-3\mathrm{i}(1+z)}}
$$
H. $$
{\frac{(1+\mathrm{i})(z+\mathrm{i})}{1+z+2\mathrm{i}(1-z)}}
$$
I. $$
{\frac{(2+\mathrm{i})(z+\mathrm{i})}{1+z-\mathrm{i}(1-z)}}
$$
J. $$
{\frac{(1+\mathrm{i})(z+\mathrm{i})}{1+\mathrm{i}z+3(1-z)}}
$$
Answer: D
To find the linear fractional transformation $w = L(z)$ such that the point $1$ is mapped to $\infty$, and the point $i$ is a double fixed point.

Options:
A. $$
\frac{z + (2 - \mathrm{i})}{(1 - \mathrm{i}) z - 1}
$$
B. $$
\frac{z - (2 \mathrm{i} + 1)}{(2 z + \mathrm{i}) - 1}
$$
C. $$
\frac{1 - (2 z + \mathrm{i})}{(2 \mathrm{i}) z + 1}
$$
D. $$
\frac{-2 z + (1 + \mathrm{i})}{(1 - \mathrm{i}) z - 2}
$$
E. $$
\frac{(1 - 2 \mathrm{i}) z - 1}{z + 1}
$$
F. $$
\frac{(z + 1)(\mathrm{i} - 2)}{1 - (z - \mathrm{i})}
$$
G. $$
\frac{(2 \mathrm{i}) z + \mathrm{i}}{\mathrm{i} - (z + 1)}
$$
H. $$
\frac{(1 + 2 \mathrm{i}) z}{z - \mathrm{i}}
$$
I. $$
\frac{(2 \mathrm{i} + 1) z - 1}{z + (1 + \mathrm{i})}
$$
J. $$
\frac{( 2 \mathrm{i}-1 ) z+1} {z-1}
$$
Answer: J
The relevant parameters of the car are as follows:
Total mass: 1600 kg;
Center of mass position: $a = 1450\, \text{mm},\, b = 1250\, \text{mm},\, h = 630\, \text{mm}$;
Engine maximum torque: $\mathrm{M_{e\,max}}= 140\, \text{N}\cdot\text{m}$;
1st gear transmission: $i_I = 0.385$;
Final drive ratio: $i_o = 4.08$;
Dynamic efficiency: $m = 0.9$;
Wheel radius: $\mathrm{r} = 3000\, \text{mm}$;
Flywheel moment of inertia: $I_f = 0.25\, \text{kg}\cdot\text{m}^2$;
Total rotational inertia of all wheels: $\Sigma I_w = 4.5\, \text{kg}\cdot\text{m}^2$ (with $I_w = 2.25\, \text{kg}\cdot\text{m}^2$ for the front wheels and $I_w = 2.25\, \text{kg}\cdot\text{m}^2$ for the rear wheels).
If we define $\frac{b}{L} \times 100\%$ as the front axle load rate, find the front axle load rate after changing the center of mass position of the original car is().

Options:
A. $$
= 6 3. 1 \%
$$
B. $$
= 5 5. 3 \%
$$
C. $$
= 6 9. 8 \%
$$
D. $$
= 5 8. 4 \%
$$
E. $$
= 6 0. 9 \%
$$
F. $$
= 5 9. 5 \%
$$
G. $$
= 7 1. 2 \%
$$
H. $$
= 6 2. 6 \%
$$
I. $$
= 6 6. 7 \%
$$
J. $$
= 7 4. 0 \%
$$
Answer: A
Land rent is the form by which the ownership of land is realized economically, and refers to the amount that must be paid for the use of all factors of production (including land, wage labor, etc.) _____.

Options:
A. Balance Sheet
B. Initial Investment
C. Variable Input
D. Equity Capital
E. Net Yield
F. Asset Value
G. Total Revenue
H. Excess Profit
I. Fixed Costs
J. Gross Income
Answer: H
The key to choosing feed is to look at the feed's ().

Options:
A. Price volatility
B. Packaging weight
C. Shelf stability
D. Seasonal demand
E. Cost-effective
F. Protein variety
G. Local availability
H. Nutritional balance
I. Caloric density
J. Digestibility rating
Answer: E
A particle with charge $q$ oscillates along the $z$-axis in a simple harmonic motion given by $z = z_{0} \mathrm{e}^{-\mathrm{i} \omega t}$. Assume $z_{0} \omega \ll c$, find its radiation field and energy flow.

Options:
A. $$
{\frac{1} {c}} e_{r} \times E_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{4} q^{2} z_{0}^{2}} {3 2 \pi^{2} c R^{2}} \operatorname{s i n}^{2} \theta e_{R}
$$
B. $$
{\frac{q} {c}} e_{\phi} \times B_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{5} q z_{0}^{2}} {8 \pi c^{2} R^{4}} \operatorname{c o s}^{2} \theta e_{\phi}
$$
C. $$
{\frac{z_{0}} {c}} e_{x} \times E_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{3} q^{2} z_{0}^{2}} {5 \pi c R^{3}} \operatorname{s i n} \theta e_{x}
$$
D. $$
{\frac{\omega z_{0}} {c^{2}}} e_{y} \times B_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{4} q^{2}} {12 \pi c^{2} R^{5}} \operatorname{s i n}^{3} \theta e_{y}
$$
E. $$
{\frac{q} {\omega}} e_{\theta} \times B_{\mathrm{e m}}
$$
$$
\frac{\mu_{0} \omega^{2} q^{3} z_{0}^{3}} {7 \pi c^{3} R} \operatorname{s i n}^{2} \theta e_{\theta}
$$
F. $$
{\frac{c} {q}} e_{\phi} \times B_{\mathrm{e m}}
$$
$$
\frac{\epsilon_{0} \omega^{2} q z_{0}^{2}} {11 \pi c^{2} R^{3}} \operatorname{s i n}^{3} \theta e_{\phi}
$$
G. $$
{\frac{\omega} {c}} e_{\theta} \times B_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{3} q^{3} z_{0}^{3}} {4 \pi c^{3} R^{3}} \operatorname{c o s}^{3} \theta e_{\theta}
$$
H. $$
{\frac{z_{0}^{2}} {c}} e_{r} \times E_{\mathrm{e m}}
$$
$$
\frac{\epsilon_{0} \omega^{5} q^{3} z_{0}} {9 \pi^{2} c^{4} R^{2}} \operatorname{c o s} \theta e_{r}
$$
I. $$
{\frac{1} {\mu_{0}}} e_{z} \times E_{\mathrm{r a d}}
$$
$$
\frac{\mu_{0} \omega^{3} q^{4} z_{0}} {10 \pi c R^{2}} \operatorname{c o s}^{2} \theta e_{z}
$$
J. $$
{\frac{1} {c^{2}}} e_{z} \times E_{\mathrm{r a d}}
$$
$$
\frac{\epsilon_{0} \omega^{2} q^{2} z_{0}^{4}} {6 \pi c R} \operatorname{s i n}^{2} \theta e_{z}
$$
Answer: A
What does the Palestrina style refer to?

Options:
A. Polyphonic madrigal
B. Secular folk song
C. Homophonic harmony
D. Renaissance opera
E. Classical sonata
F. Counterpoint melodies
G. A cappella style
H. Sacred symphony
I. Gregorian chant
J. Baroque texture
Answer: G
What is the predecessor of the Baroque oboe?

Options:
A. Clarinet
B. Bassoon
C. Serpent
D. Shawms
E. Flute
F. Oboe
G. Recorder
H. Crumhorn
I. Bagpipe
J. Dulcian
Answer: F
What third-tier characters did Mozart portray in the opera *The Marriage of Figaro*?

Options:
A. The Innkeeper Bartolo and the Florist Caterina.
B. The Gardener Antonio and the Washerwoman Rosina.
C. The Tailor Antonio and the Maid Cherubina.
D. The Farmer Figaro and the Cook Marie.
E. The Shepherd Figaro and the Nurse Margarita.
F. The Butler Basilio and the Seamstress Fanchette.
G. The Miller Marcel and the Housekeeper Suzette.
H. The Valet Basilio and the Governess Bianca.
I. The Merchant Figaro and the Dancer Lucia.
J. The Barber Figaro and the Maid Suzanne.
Answer: J
What are the important representative works of French lyrical opera?

Options:
A. Rameau's *Platée*; Gounod's *The Queen of Sheba*; Thomas's *La Cour de Célimène*; Saint-Saëns's *Les Barbares*.
B. Gounod's *Romeo and Juliet*; Bizet's *Carmen*; Delibes's *Lakmé*; Rameau's *Hippolyte et Aricie*.
C. Ravel's *L'Enfant et les Sortilèges*; Berlioz's *Benvenuto Cellini*; Debussy's *Le Martyre de saint Sébastien*; Offenbach's *Orpheus in the Underworld*.
D. Offenbach's *The Tales of Hoffmann*; Charpentier's *Médée*; Gluck's *Orfeo ed Euridice*; Massenet's *Thaïs*.
E. Berlioz's *Les Troyens*; Debussy's *Pelléas et Mélisande*; Lully's *Armide*; Meyerbeer's *Les Huguenots*.
F. Massenet's *Manon*; Meyerbeer's *Robert le Diable*; Charpentier's *La Descente d'Orphée aux Enfers*; Lully's *Atys*.
G. Bizet's *Ivan IV*; Massenet's *Le Cid*; Debussy's *La Chute de la Maison Usher*; Lully's *Cadmus et Hermione*.
H. Delibes's *Coppelia*; Meyerbeer's *Dinorah*; Charpentier's *David et Jonathas*; Offenbach's *Fantasio*.
I. Gluck's *Alceste*; Saint-Saëns's *Ascanio*; Bizet's *Djamileh*; Thomas's *Mignon*.
J. Gounod's *Faust*; Thomas's *Hamlet*; Bizet's *The Pearl Fishers*; Saint-Saëns's *Samson and Delilah*.
Answer: J
What are the differences in composition between Webern and Schoenberg?

Options:
A. Concise.
B. Ornamental.
C. Repetitive.
D. Fragmented.
E. Predictable.
F. Chaotic.
G. Conventional.
H. Melodic.
I. Simplistic.
J. Lengthy.
Answer: A
What musical established Broadway as the center of American musical theater production?

Options:
A. Wicked
B. Rent
C. Phantom of the Opera
D. Cats
E. Chicago
F. West Side Story
G. Hamilton
H. The Sound of Music
I. Les Misérables
J. Miss Saigon
Answer: H
A star is observed to cross the meridian at an elevation of $6 7^{\circ}$ , as seen from an observatory at a latitude of $5 2^{\circ}$ north.
What would be the declination of a star observed to transit at an elevation of 20°?

Options:
A. $- 1 6^{\circ}$
B. $- 2 4^{\circ}$
C. $- 1 5^{\circ}$
D. $- 2 3^{\circ}$
E. $- 1 7^{\circ}$
F. $- 2 2^{\circ}$
G. $- 2 0^{\circ}$
H. $- 2 1^{\circ}$
I. $- 1 8^{\circ}$
J. $- 1 9^{\circ}$
Answer: I
A star has a luminosity of $8 \times1 0^{2 6} \mathrm{W}$ and a diameter of $8 \times1 0^{8} \mathrm{m}$ . Use the Stefan-Boltzmann Law to estimate its surface temperature.

Options:
A. $$
9. 6 2 5 \times1 0^{3} \mathrm{K}
$$
B. $$
7. 2 1 5 \times1 0^{3} \mathrm{K}
$$
C. $$
12. 4 8 7 \times1 0^{3} \mathrm{K}
$$
D. $$
6. 5 7 2 \times1 0^{3} \mathrm{K}
$$
E. $$
5. 4 9 3 \times1 0^{3} \mathrm{K}
$$
F. $$
10. 9 8 5 \times1 0^{3} \mathrm{K}
$$
G. $$
9. 1 5 2 \times1 0^{3} \mathrm{K}
$$
H. $$
8. 3 4 6 \times1 0^{3} \mathrm{K}
$$
I. $$
8. 7 1 4 \times1 0^{3} \mathrm{K}
$$
J. $$
11. 3 6 8 \times1 0^{3} \mathrm{K}
$$
Answer: G
Show that the surface temperature, $T,$ of a rotating spherical body at a distance $R$ from the Sun falls as the inverse square root of $R$ .
Knowing that a spacecraft in Earth orbit prior to orbital insertion towards the Sun reached a stable temperature of 260K, how close (in AU) could this spacecraft be allowed to approach the Sun if it could withstand a surface temperature of 1000K?

Options:
A. 0.0547AU
B. 0.0593AU
C. 0.0773AU
D. 0.0719AU
E. 0.0882AU
F. 0.0636AU
G. 0.0821AU
H. 0.0845AU
I. 0.0908AU
J. 0.0654AU
Answer: C
Stellar magnitudes obey a logarithmic scale with the difference in brightness between a magnitude 1 star and a magnitude 6 star defined to be exactly 100. Given that the eye has a dark adapted pupil size of 5 mm, how large would a telescope aperture have to be in order to observe a star 7 magnitudes fainter than the human eye can detect?

Options:
A. 115.8
B. 130.2
C. 125.6
D. 100.2
E. 105.6
F. 120.4
G. 145.5
H. 140.3
I. 135.9
J. 110.7
Answer: C
Calculate the resolution of the 2.4-m mirror of the Hubble Space Telescope at a wavelength of $0. 5 \times1 0^{-6} \mathrm{m}$ . How large would an array of radio
telescopes operating at a wavelength of 0.06 m need to be to equal this
resolution?

Options:
A. $$
2 . 8 0 \times1 0^{5} \mathrm{m}
$$
B. $$
3 . 2 0 \times1 0^{5} \mathrm{m}
$$
C. $$
2 . 9 0 \times1 0^{5} \mathrm{m}
$$
D. $$
3 . 1 5 \times1 0^{5} \mathrm{m}
$$
E. $$
2. 8 8 \times1 0^{5} \mathrm{m}
$$
F. $$
3 . 0 5 \times1 0^{5} \mathrm{m}
$$
G. $$
2 . 5 4 \times1 0^{5} \mathrm{m}
$$
H. $$
2 . 6 5 \times1 0^{5} \mathrm{m}
$$
I. $$
2 . 9 9 \times1 0^{5} \mathrm{m}
$$
J. $$
2 . 7 0 \times1 0^{5} \mathrm{m}
$$
Answer: E
$\mathrm{A}$ star has a surface temperature twice that of our Sun. The star has a mass 3 times that of our Sun and has a similar density. Calculate how much more energy it radiates relative to the Sun and estimate its lifetime compared with the Sun.

Options:
A. $$33.28, 0.33$$
B. $$33.28, 0.45$$
C. $$40.11, 0.05$$
D. $$28.32, 0.12$$
E. $$33.28, 0.09$$
F. $$12.09, 0.18$$
G. $$50.78, 0.07$$
H. $$25.76, 0.19$$
I. $$45.63, 0.25$$
J. $$22.15, 0.15$$
Answer: E
A neutron star has a radius of 11.5 km and a mass of about $1.35M_{sun}$. Calculate the density of this neutron star in kilograms per cubic centimetre.

Options:
A. $$
4. 2 \times1 0^{11} \mathrm{kg \cdot cm}^{-3}
$$
B. $$ 3. 4 \times1 0^{11} \mathrm{kg \cdot cm}^{-3} $$
C. $$ 8. 3 \times1 0^{10} \mathrm{kg \cdot cm}^{-3} $$
D. $$ 3. 8 \times1 0^{10} \mathrm{kg \cdot cm}^{-3} $$
E. $$ 5. 0 \times1 0^{12} \mathrm{kg \cdot cm}^{-3} $$
F. $$ 1. 7 \times1 0^{12} \mathrm{kg \cdot cm}^{-3} $$
G. $$ 5. 6 \times1 0^{11} \mathrm{kg \cdot cm}^{-3} $$
H. $$ 6. 9 \times1 0^{10} \mathrm{kg \cdot cm}^{-3} $$
I. $$ 2. 4 \times1 0^{12} \mathrm{kg \cdot cm}^{-3} $$
J. $$ 9. 1 \times1 0^{10} \mathrm{kg \cdot cm}^{-3} $$
Answer: A
The wavelength of the spectral line for the transition of sodium atoms from $3^2P_{1/2}→ 3^2S_{1/2}$ is 589.6 nm. In a magnetic field of $B = 2.5T$, Zeeman splitting occurs. Observing from a direction perpendicular to the magnetic field, how many spectral lines does it split into? And calculate the wavelength of each spectral line.

Options:
A. $$
5 lines. 589.520 nm, 589.550 nm, 589.580 nm, 589.610 nm, 589.640 nm.
$$
B. $$
3 lines. 589.540 nm, 589.600 nm, 589.650 nm.
$$
C. $$
3 lines. 589.560 nm, 589.600 nm, 589.640 nm.
$$
D. $$
6 lines. 589.530 nm, 589.550 nm, 589.570 nm, 589.590 nm, 589.610 nm, 589.630 nm.
$$
E. $$
5 lines. 589.545 nm, 589.565 nm, 589.585 nm, 589.605 nm, 589.625 nm.
$$
F. $$
2 lines. 589.570 nm, 589.630 nm.
$$
G. $$
3 lines. 589.550 nm, 589.585 nm, 589.620 nm.
$$
H. $$
4 lines. 589.546 nm, 589.573 nm, 589.627 nm, 589.654 nm.
$$
I. $$
1 line. 589.600 nm.
$$
J. $$
2 lines. 589.540 nm, 589.660 nm.
$$
Answer: H
The bond length of the $^{12}C^{18}O$ molecule is $R_{0}=0.1128 \mathrm{nm}$.
If the Raman scattering of this molecule is measured, calculate the spacing between the small Raman shift spectral lines and the wavelength difference between the first anti-Stokes line and the first Stokes line is _____.

Options:
A. $$
24.50cm^{-1} or 24.55cm^{-1}
$$
B. $$
22.08cm^{-1} or 22.10cm^{-1}
$$
C. $$
23.08cm^{-1} or 23.10cm^{-1}
$$
D. $$
20.05cm^{-1} or 20.10cm^{-1}
$$
E. $$
25.08cm^{-1} or 25.15cm^{-1}
$$
F. $$
19.50cm^{-1} or 19.55cm^{-1}
$$
G. $$
18.08cm^{-1} or 18.10cm^{-1}
$$
H. $$
20.08cm^{-1} or 20.10cm^{-1}
$$
I. $$
26.08cm^{-1} or 26.10cm^{-1}
$$
J. $$
21.50cm^{-1} or 21.55cm^{-1}
$$
Answer: B
When an atom of K (Z=19) transitions from the first excited state to the ground state, it produces two fine structure spectral lines with wavelengths of 7664 ${\AA}$ and 7699 ${\AA}$. If the atom is placed in a magnetic field $B$, the spectral lines will further split. To make the energy gap between the highest and lowest levels after the splitting 1.5 times the original energy gap, what should the strength of the applied magnetic field $B$ be?

Options:
A. $$
1.8 \times 10^{11}T
$$
B. $$
3.7 \times 10^{10}T
$$
C. $$
5.4 \times 10^{10}T
$$
D. $$
6.8 \times 10^{9}T
$$
E. $$
3.7 \times 10^{9}T
$$
F. $$
2.1 \times 10^{11}T
$$
G. $$
2.5 \times 10^{10}T
$$
H. $$
7.3 \times 10^{10}T
$$
I. $$
4.5 \times 10^{10}T
$$
J. $$
3.9 \times 10^{10}T
$$
Answer: B
Light with a wavelength of 2000 ${\AA}$ illuminates the surface of aluminum. Given that the work function of aluminum is $4.2 \mathrm{eV}$, what is the stopping voltage of the photoelectric effect?

Options:
A. 5.2V
B. 3V
C. 1.5V
D. 2V
E. 4.5V
F. 2.8V
G. 7.5V
H. 0.8V
I. 6V
J. 1.2V
Answer: D
$\frac{e}{m}$ is the charge-to-mass ratio of an electron. $L$, $S$, and $J$ represent the total orbital angular momentum, total spin angular momentum, and total angular momentum of the electron, respectively. What is the total magnetic moment of the atom?

Options:
A. $$
- \frac{e} {2 m} ( L-J ) \,
$$
B. $$
\frac{e} {m} ( L+J ) \,
$$
C. $$
\frac{e} {2 m} ( J-S ) \,
$$
D. $$
- \frac{e} {2 m} ( L-S ) \,
$$
E. $$
\frac{e} {2 m} ( L+S ) \,
$$
F. $$
- \frac{e} {2 m} ( J+S ) \,
$$
G. $$
\frac{e} {m} ( J+S ) \,
$$
H. $$
- \frac{e} {m} ( J-S ) \,
$$
I. $$
- \frac{e} {m} ( L+S ) \,
$$
J. $$
- \frac{e} {2 m} ( L+J ) \,
$$
Answer: F
What is the Rydberg equation for He$^{+}$ ions? (Where $R$ is the Rydberg constant, and $m, n$ are integers).

Options:
A. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 2 )^{3}}-\frac{1} {( n )^{3}} ]
$$
B. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 2 )^{3}}-\frac{1} {( n / 2 )^{3}} ]
$$
C. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( 2m )^{2}}-\frac{1} {( n )^{2}} ]
$$
D. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m^{2} )}-\frac{1} {( n^{3} / 2 )} ]
$$
E. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 2 )^{2}}-\frac{1} {( n / 2 )^{2}} ]
$$
F. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 2 )}-\frac{1} {( n / 3 )^{2}} ]
$$
G. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m )}-\frac{1} {( n / 4 )^{2}} ]
$$
H. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m )^{2}}-\frac{1} {( 2n )^{2}} ]
$$
I. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 3 )^{2}}-\frac{1} {( n / 2 )^{3}} ]
$$
J. $$
{\widetilde{\nu}}=\mathrm{R} [ \frac{1} {( m / 4 )^{2}}-\frac{1} {( n / 2 )^{2}} ]
$$
Answer: E
The wavelength of the $K_{\alpha}$ line of the $X$-ray emitted by a certain anode target is $0.1949 \mathrm{nm}$. What is the atomic number of the target material?

Options:
A. 18
B. 24
C. 36
D. 20
E. 28
F. 32
G. 34
H. 26
I. 30
J. 22
Answer: H
If the charge values of the proton and electron in a hydrogen atom are both doubled, by what factor will the energy of the photon emitted during the transition from $n=2$ to $n=1$ be multiplied?

Options:
A. 16
B. 128
C. 4
D. 32
E. 1
F. 64
G. 256
H. 8
I. 512
J. 2
Answer: A
When a germanium crystal is cooled from 600°C to 300°C, the equilibrium concentration of Schottky defects (vacancies) in the germanium crystal decreases by six orders of magnitude. (The gas constant $R = 8.314 \, {J}/({mol·K})$.) Calculate the formation energy of the Schottky defects in the germanium crystal is ().

Options:
A. $1.83\times10^{5}J/mol$
B. $3.20\times10^{5}J/mol$
C. $5.10\times10^{5}J/mol$
D. $2.83\times10^{5}J/mol$
E. $3.83\times10^{5}J/mol$
F. $4.00\times10^{5}J/mol$
G. $2.75\times10^{5}J/mol$
H. $3.50\times10^{5}J/mol$
I. $4.90\times10^{5}J/mol$
J. $3.95\times10^{5}J/mol$
Answer: E
A certain metal has a body-centered cubic structure with a unit cell edge length of 0.286 nm and a density of 7.88 g/cm³. Calculate the relative atomic mass of this metal is ().

Options:
A. 50.92
B. 53.67
C. 58.12
D. 57.99
E. 52.45
F. 60.88
G. 55.49
H. 56.33
I. 59.75
J. 54.21
Answer: G
Lithium oxide ($Li_2O$) has an anti-fluorite structure. The ionic radius of a lithium ion is known to be 0.060 nm, and the ionic radius of an oxygen ion is 0.140 nm; the atomic mass of oxygen ($O$) is 16, and the atomic mass of lithium ($Li$) is 6.94. Calculate the lattice parameter and the density of the $Li_2O$ crystal is ().

Options:
A. 0.410nm,3.100g/cm³
B. 0.430nm,3.015g/cm³
C. 0.420nm,3.050g/cm³
D. 0.400nm,3.013g/cm³
E. 0.405nm,3.000g/cm³
F. 0.390nm,2.980g/cm³
G. 0.380nm,2.950g/cm³
H. 0.425nm,3.250g/cm³
I. 0.375nm,3.120g/cm³
J. 0.395nm,3.200g/cm³
Answer: D
In horizontal motion, Coriolis deviation can be divided into three components for assessment. One is (), judged by the three-hour pressure change; another is (), judged by the divergence and convergence of the isobars (or contour lines); and the third is (), judged by the curvature of the isobars (or contour lines).

Options:
A. Transfer wind, lateral rotational convergence, longitudinal isobar rotation
B. Transfer contour, lateral divergence deviation, longitudinal pressure rotation
C. Transform wind, lateral isobar deviation, longitudinal contour deviation
D. Transformer drift, rotational convergence deviation, divergence rotational deviation
E. Transition flow, lateral deviation pressure, longitudinal curvature pressure
F. Transitional wind, lateral contour divergence, vertical rotational deviation
G. Transformer wind, lateral rotational deviation, longitudinal rotational deviation
H. Transformational wind, rotational contour deviation, axial divergence deviation
I. Transversal wind, radial rotational deviation, axial rotational deviation
J. Transmission drift, curvature rotation, isobar divergence variation
Answer: G
Assuming that an average of one coronal mass ejection occurs per day and that a typical CME ejects $1 0^{1 3}$ kg of material, estimate the annual mass loss from CMEs and compare your answer with the annual mass loss from the solar wind. Express your answer as a percentage of CME mass loss to solar wind mass loss.

Options:
A. 7%
B. 5%
C. 11%
D. 8%
E. 6%
F. 10%
G. 3%
H. 14%
I. 12%
J. 2%
Answer: A
Make a rough estimate of how long it will take for Earth's rotation period to reach $4 7$ days, at which time it will be synchronized with the Moon's orbital period.

Options:
A. $$
1. 5 \times1 0^{1 0} \mathrm{~ y r}
$$
B. $$
7. 0 \times1 0^{1 0} \mathrm{~ y r}
$$
C. $$
6. 9 \times1 0^{1 2} \mathrm{~ y r}
$$
D. $$
5. 7 \times1 0^{1 2} \mathrm{~ y r}
$$
E. $$
2. 4 \times1 0^{1 1} \mathrm{~ y r}
$$
F. $$
8. 6 \times1 0^{1 1} \mathrm{~ y r}
$$
G. $$
1. 2 \times1 0^{1 3} \mathrm{~ y r}
$$
H. $$
4. 3 \times1 0^{8} \mathrm{~ y r}
$$
I. $$
3. 8 \times1 0^{1 1} \mathrm{~ y r}
$$
J. $$
9. 1 \times1 0^{9} \mathrm{~ y r}
$$
Answer: E
Based on the observed rate at which North America and Eurasia are separating from each other, when were the two continents joined together as Laurasia? Assume that the Atlantic Ocean is roughly 4800 km (3000 miles) wide.

Options:
A. $$
1. 6 \times1 0^{8} \mathrm{~ y r}
$$
B. $$
1. 15 \times1 0^{8} \mathrm{~ y r}
$$
C. $$
1. 13 \times1 0^{8} \mathrm{~ y r}
$$
D. $$
1. 9 \times1 0^{8} \mathrm{~ y r}
$$
E. $$
1. 17 \times1 0^{8} \mathrm{~ y r}
$$
F. $$
1. 0 \times1 0^{8} \mathrm{~ y r}
$$
G. $$
1. 4 \times1 0^{8} \mathrm{~ y r}
$$
H. $$
2. 5 \times1 0^{8} \mathrm{~ y r}
$$
I. $$
1. 18 \times1 0^{8} \mathrm{~ y r}
$$
J. $$
2. 0 \times1 0^{8} \mathrm{~ y r}
$$
Answer: A
Assuming that the two moons are in circular orbits, determine the orbital radii of Phobos and Deimos. Express your answers in units of the radius of Mars.

Options:
A. $$
2. 7 5 \; R_{\mathrm{M a r s}}
$$
$$
6. 8 9 \ R_{\mathrm{M a r s}}.
$$
B. $$
3. 0 0 \; R_{\mathrm{M a r s}}
$$
$$
7. 1 5 \ R_{\mathrm{M a r s}}.
$$
C. $$
3. 1 5 \; R_{\mathrm{M a r s}}
$$
$$
7. 0 1 \ R_{\mathrm{M a r s}}.
$$
D. $$
2. 7 1 \; R_{\mathrm{M a r s}}
$$
$$
6. 8 5 \ R_{\mathrm{M a r s}}.
$$
E. $$
2. 5 5 \; R_{\mathrm{M a r s}}
$$
$$
6. 9 0 \ R_{\mathrm{M a r s}}.
$$
F. $$
2. 8 4 \; R_{\mathrm{M a r s}}
$$
$$
6. 9 5 \ R_{\mathrm{M a r s}}.
$$
G. $$
2. 6 0 \; R_{\mathrm{M a r s}}
$$
$$
6. 7 5 \ R_{\mathrm{M a r s}}.
$$
H. $$
3. 2 5 \; R_{\mathrm{M a r s}}
$$
$$
7. 5 0 \ R_{\mathrm{M a r s}}.
$$
I. $$
2. 6 8 \; R_{\mathrm{M a r s}}
$$
$$
6. 7 9 \ R_{\mathrm{M a r s}}.
$$
J. $$
2. 9 2 \; R_{\mathrm{M a r s}}
$$
$$
7. 2 0 \ R_{\mathrm{M a r s}}.
$$
Answer: A
The mass and radius of Miranda are $8 \times1 0^{1 9}$ kg and 236 km, respectively. What is the escape velocity from the surface of Miranda?

Options:
A. $$
0. 4 2 2 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
B. $$
0. 1 9 4 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
C. $$
0. 2 5 5 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
D. $$
0. 0 7 7 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
E. $$
0. 5 6 4 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
F. $$
0. 1 5 2 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
G. $$
0. 2 9 9 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
H. $$
0. 3 1 7 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
I. $$
0. 0 9 8 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
J. $$
0. 2 1 3 \; \mathrm{k m} \; \mathrm{s}^{-1}.
$$
Answer: J
In this problem you will estimate the amount of energy released per second by the radioactive decay of $\mathrm{^{2 6}_{1 3} A l}$ inside Vesta during its lifetime.
Vesta has a radius of 250 km and a density of 2900 kg $\mathrm{m}^{-3}$ . Assuming spherical symmetry,
estimate the asteroid's mass.

Options:
A. $$
2. 3 \times1 0^{2 2} \ \mathrm{k g}.
$$
B. $$
7. 1 \times1 0^{1 8} \ \mathrm{k g}.
$$
C. $$
6. 7 \times1 0^{1 8} \ \mathrm{k g}.
$$
D. $$
4. 8 \times1 0^{1 9} \ \mathrm{k g}.
$$
E. $$
8. 2 \times1 0^{2 1} \ \mathrm{k g}.
$$
F. $$
3. 9 \times1 0^{1 8} \ \mathrm{k g}.
$$
G. $$
9. 4 \times1 0^{2 1} \ \mathrm{k g}.
$$
H. $$
5. 6 \times1 0^{2 0} \ \mathrm{k g}.
$$
I. $$
3. 5 \times1 0^{1 9} \ \mathrm{k g}.
$$
J. $$
1. 9 \times1 0^{2 0} \ \mathrm{k g}.
$$
Answer: J
HD 63454 is a K4V star known to have an extrasolar planet orbiting it in a circular orbit with an orbital period of 2.81782 d, producing a maximum radial reflex velocity of 64 3 m $\mathrm{S}^{-1}$ relative to the center of mass of the star. The distance to HD 63454 is 35.80 pc. Consulting Appendix G, determine
the semimajor axis of the planet's orbit.

Options:
A. $$
5. 5 \times1 0^{8} \mathrm{m}.
$$
B. $$
6. 5 \times1 0^{9} \mathrm{m}.
$$
C. $$
5. 2 \times1 0^{9} \mathrm{m}.
$$
D. $$
7. 8 \times1 0^{7} \mathrm{m}.
$$
E. $$
1. 7 \times1 0^{9} \mathrm{m}.
$$
F. $$
8. 9 \times1 0^{8} \mathrm{m}.
$$
G. $$
2. 3 \times1 0^{8} \mathrm{m}.
$$
H. $$
4. 1 \times1 0^{9} \mathrm{m}.
$$
I. $$
3. 4 \times1 0^{8} \mathrm{m}.
$$
J. $$
9. 6 \times1 0^{7} \mathrm{m}.
$$
Answer: C
The refractive indices of the uniform optical fiber core and cladding are $n_{1} = 1.50 \,, \, \, n_{2} = 1.45$. Calculate the maximum time delay difference $\Delta\tau_{\operatorname*{max}}$ caused by the optical path difference of meridional rays in a 1m-long optical fiber $\pm$. The result is _____.

Options:
A. 0.22ns
B. 0.18ns
C. 0.11ns
D. 0.12ns
E. 0.10ns
F. 0.17ns
G. 0.14ns
H. 0.16ns
I. 0.20ns
J. 0.19ns
Answer: H
Step index fiber core radius $a = \! 25\mu{m}$, refractive index $n_{1}=1.5$, relative refractive index difference $\Delta \! = \! 1\%$, length $L=1$km. Problem: If the cladding and surface coating of the fiber are removed, the NA (numerical aperture) and maximum delay difference of the bare fiber are calculated as ____.

Options:
A. $$
1.6; 1.7 \ \mu\mathrm{s}
$$
B. $$
0.9; 1.3 \ \mu\mathrm{s}
$$
C. $$
1.3; 1.5 \ \mu\mathrm{s}
$$
D. $$
1.5; 1.6 \ \mu\mathrm{s}
$$
E. $$
1.4; 1.0 \ \mu\mathrm{s}
$$
F. $$
1.2; 1.4 \ \mu\mathrm{s}
$$
G. $$
1.1; 0.8 \ \mu\mathrm{s}
$$
H. $$
1; 1. 1 \ \mu\mathrm{s}
$$
I. $$
0.8; 1.2 \ \mu\mathrm{s}
$$
J. $$
0.7; 0.9 \ \mu\mathrm{s}
$$
Answer: H
A certain laser uses GaAs as the active medium. What are the frequency and wavelength of the light it emits?

Options:
A. $$
0. 9 3 \mu\mathrm{m}
$$
$$
3. 2 1 1 \times 1 0^{1 4} \mathrm{~ H z}
$$
B. $$
0. 7 2 \mu\mathrm{m}
$$
$$
3. 9 6 2 \times 1 0^{1 4} \mathrm{~ H z}
$$
C. $$
0. 6 5 \mu\mathrm{m}
$$
$$
4. 6 1 5 \times 1 0^{1 4} \mathrm{~ H z}
$$
D. $$
0. 5 4 \mu\mathrm{m}
$$
$$
4. 8 0 3 \times 1 0^{1 4} \mathrm{~ H z}
$$
E. $$
1. 5 5 \mu\mathrm{m}
$$
$$
1. 9 3 5 \times 1 0^{1 4} \mathrm{~ H z}
$$
F. $$
1. 4 8 \mu\mathrm{m}
$$
$$
2. 0 1 0 \times 1 0^{1 4} \mathrm{~ H z}
$$
G. $$
0. 8 7 \mu\mathrm{m}
$$
$$
3. 4 4 5 \times1 0^{1 4} \mathrm{~ H z}
$$
H. $$
1. 0 6 \mu\mathrm{m}
$$
$$
2. 8 3 1 \times 1 0^{1 4} \mathrm{~ H z}
$$
I. $$
1. 0 0 \mu\mathrm{m}
$$
$$
3. 0 0 0 \times 1 0^{1 4} \mathrm{~ H z}
$$
J. $$
1. 2 0 \mu\mathrm{m}
$$
$$
2. 5 0 0 \times 1 0^{1 4} \mathrm{~ H z}
$$
Answer: G
A GaAs PIN photodiode produces, on average, one electron hole pair for every three incident photons. Let's say all the electrons are collected. The received power in the 0.8um band is $10 ^{-7} \mathbf{w}$, what is the average output photocurrent?

Options:
A. $$35 nA$$
B. $$12 nA$$
C. $$25 nA$$
D. $$22 nA$$
E. $$18 nA$$
F. $$30 nA$$
G. $$40 nA$$
H. $$28 nA$$
I. $$20 nA$$
J. $$15 nA$$
Answer: D
For a 140 Mb/s digital optical fiber communication system, the transmitted optical power is -3 dBm, the receiver sensitivity is -38 dBm, the system margin is 4 dB, the connector loss is 0.5 dB per pair, the average splice loss is 0.05 dB/km, the optical fiber attenuation coefficient is 0.4 dB/km, and the fiber loss margin is 0.05 dB/km. Calculate the repeater distance $L$ as _____.

Options:
A. 70km
B. 75km
C. 85km
D. 65km
E. 60km
F. 50km
G. 45km
H. 90km
I. 55km
J. 80km
Answer: E
PN photodiode at 0.85 $\mu\mathrm{m}$ wavelength quantum efficiency is 0.7, try to calculate its responsiveness, and incident light power of -40 dBm photocurrent is what?

Options:
A. $$
\approx0. 2 9 \mathrm{~ A} / \mathbf{W}
$$
$$
2. 9 \times1 0^{-9} \mathrm{~ A}
$$
B. $$
\approx0. 3 0 \mathrm{~ A} / \mathbf{W}
$$
$$
3. 0 \times1 0^{-8} \mathrm{~ A}
$$
C. $$
\approx0. 4 9 \mathrm{~ A} / \mathbf{W}
$$
$$
4. 9 \times1 0^{-8} \mathrm{~ A}
$$
D. $$
\approx0. 4 3 \mathrm{~ A} / \mathbf{W}
$$
$$
4. 3 \times1 0^{-7} \mathrm{~ A}
$$
E. $$
\approx0. 4 8 \mathrm{~ A} / \mathbf{W}
$$
$$
4. 8 \times1 0^{-8} \mathrm{~ A}
$$
F. $$
\approx0. 3 7 \mathrm{~ A} / \mathbf{W}
$$
$$
3. 7 \times1 0^{-8} \mathrm{~ A}
$$
G. $$
\approx0. 6 2 \mathrm{~ A} / \mathbf{W}
$$
$$
6. 2 \times1 0^{-8} \mathrm{~ A}
$$
H. $$
\approx0. 7 0 \mathrm{~ A} / \mathbf{W}
$$
$$
7. 0 \times1 0^{-8} \mathrm{~ A}
$$
I. $$
\approx0. 5 4 \mathrm{~ A} / \mathbf{W}
$$
$$
5. 4 \times1 0^{-8} \mathrm{~ A}
$$
J. $$
0. 1 5 \mathrm{~ A} / \mathbf{W}
$$
$$
1. 5 \times1 0^{-8} \mathrm{~ A}
$$
Answer: E
A step-index multimode optical fiber with a numerical aperture of 0.2 can support 882 transmission modes at a wavelength of 850 nm. What is the diameter of its core?

Options:
A. $$
2 5. 6 \mathrm{~} \mu\mathrm{m}
$$
B. $$
3 3. 7 \mathrm{~} \mu\mathrm{m}
$$
C. $$
2 7. 8 \mathrm{~} \mu\mathrm{m}
$$
D. $$
2 9. 1 \mathrm{~} \mu\mathrm{m}
$$
E. $$
3 0. 2 \mathrm{~} \mu\mathrm{m}
$$
F. $$
3 2. 0 \mathrm{~} \mu\mathrm{m}
$$
G. $$
2 8. 4 \mathrm{~} \mu\mathrm{m}
$$
H. $$
2 8. 9 \mathrm{~} \mu\mathrm{m}
$$
I. $$
3 1. 4 \mathrm{~} \mu\mathrm{m}
$$
J. $$
2 6. 5 \mathrm{~} \mu\mathrm{m}
$$
Answer: G
It is known that the central wavelength of the GaAs laser diode is $0.85 \mu. \mathrm{m}$, the resonator length is 0.4mm, and the material refractive index is 3.7. If the optical gain of the laser is always greater than the total attenuation of the resonator in the range of $0.8 0 \mu. \mathrm{m} \leq\lambda\ leq0.9 $, the number of longitudinal modes that can be excited in the laser is ____.

Options:
A. 399
B. 418
C. 425
D. 415
E. 410
F. 405
G. 408
H. 420
I. 402
J. 412
Answer: E
A certain $\delta$ optical pulse signal becomes broadened into a Gaussian waveform after being transmitted through a 50 km multimode optical fiber. Given that its root mean square broadening $\sigma = 2.14 \times 10^{-8}$ s, what is the bandwidth of the multimode optical fiber per kilometer? (Take $\gamma = 0.8$)

Options:
A. $$
2 5 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
B. $$
1 8 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
C. $$
1 9 5 \mathrm{M H z} \cdot\mathrm{k m}
$$
D. $$
1 7 5 \mathrm{M H z} \cdot\mathrm{k m}
$$
E. $$
1 6 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
F. $$
2 1 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
G. $$
2 3 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
H. $$
2 0 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
I. $$
2 2 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
J. $$
2 4 0 \mathrm{M H z} \cdot\mathrm{k m}
$$
Answer: H
To manufacture a step-index multimode optical fiber with a quartz core, given the normalized frequency $V=7.5$, numerical aperture $\mathbf{N}\mathbf{A}=0.30$, operating wavelength $\lambda=0.85 \mu\mathrm{m}$, and $n_{1}=1.45$, what is the refractive index of the cladding $n_{2}$? What should be the radius of the fiber core?

Options:
A. $$
\approx1. 4 2
$$
$$
\approx2 9 \, \mu\mathrm{m}
$$
B. $$
\approx1. 4 1
$$
$$
\approx4 2 \, \mu\mathrm{m}
$$
C. $$
\approx1. 4 5
$$
$$
\approx2 6 \, \mu\mathrm{m}
$$
D. $$
\approx1. 4 4
$$
$$
\approx2 8 \, \mu\mathrm{m}
$$
E. $$
\approx1. 3 9
$$
$$
\approx3 5 \, \mu\mathrm{m}
$$
F. $$
\approx1. 4 3
$$
$$
\approx3 8 \, \mu\mathrm{m}
$$
G. $$
\approx1. 3 8
$$
$$
\approx4 0 \, \mu\mathrm{m}
$$
H. $$
\approx1. 4 2
$$
$$
\approx3 4 \, \mu\mathrm{m}
$$
I. $$
\approx1. 4 3
$$
$$
\approx3 2 \, \mu\mathrm{m}
$$
J. $$
\approx1. 4 0
$$
$$
\approx3 6 \, \mu\mathrm{m}
$$
Answer: H
A 1.3 pm optical wave system utilizes a 50 km optical fiber link, where the receiver requires a minimum power of 0.3 $\mu\mathbf{W}$. The fiber attenuation is 0.5 dB/km, and the connection loss is 0.2 dB. At both ends, there is an active connector with a loss of 1 dB each. Calculate the minimum power required to be emitted into the optical fiber is().

Options:
A. 0.172mW
B. 0.183mW
C. 0.157mW
D. 0.125mW
E. 0.139mW
F. 0.146mW
G. 0.200mW
H. 0.180mW
I. 0.191mW
J. 0.165mW
Answer: C
The circular ring-shaped iron core has an inner radius of $R_{1}$, an outer radius of $R_{2}$, a thickness of $h$, and a permeability of $p$ $( \mu >> \mu_{0} )$. It is uniformly and tightly wound with $N$ turns of coil, and the current in the coil is $I$. The magnetic induction strength and magnetic flux in the iron core, as well as the magnetic linkage of the coil is ().

Options:
A. Magnetic flux density: $B={\frac{\mu_{0} h I} {2}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N h} {2 }} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N^{2} h^{2}} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
B. Magnetic flux density: $B={\frac{\mu_{0} I N R_{1}} {h}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I h N^{2}} {4 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
C. Magnetic flux density: $B={\frac{\mu_{0} I N h} {2 \pi r R_{2}}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N^{2}} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I h N^{2}} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
D. Magnetic flux density: $B={\frac{\mu_{0} I N} {2 \pi}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N^{2} h^{2}} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
E. Magnetic flux density: $B={\frac{\mu I N} {2 r}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu I N h} {2}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu I N^{2} h} {2} \operatorname{l n} \frac{R_{2}} {R_{1}}$
F. Magnetic flux density: $B={\frac{\mu_{0} I N} {2 \pi r}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N h} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N^{2} h} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
G. Magnetic flux density: $B={\frac{\mu_{0} I N R_{1}} {2 h}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N h R_{2}} {2 \pi}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N R_{1} h} {2 \pi}$
H. Magnetic flux density: $B={\frac{\mu_{0} I N} {r}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N h} {4 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N^{2} R_{1}} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
I. Magnetic flux density: $B={\frac{\mu_{0} I R_{2}} {2 \pi h}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N h^{2}} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N^{2} h} {4 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
J. Magnetic flux density: $B={\frac{\mu_{0} I N^{2}} {2 \pi r}} e_{\alpha}$
Magnetic flux: $\phi={\frac{\mu_{0} I N R_{2}} {2 \pi}} \operatorname{l n} {\frac{R_{2}} {R_{1}}}$
Magnetic linkage: $\psi=\frac{\mu_{0} I N h} {2 \pi} \operatorname{l n} \frac{R_{2}} {R_{1}}$
Answer: F
The electrodes consist of two large parallel plates, with air as the dielectric between them. The voltage between the two plates is constant. When the plates move towards each other at a constant velocity $\nu$ along the normal direction of the plane of the plates, what is the displacement current density between the plates?

Options:
A. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu} {\left( x_{0}-\nu t \right)^{2}}} \right]} \boldsymbol{e}_{x}
$$
B. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu} {\left( x_{0}+\nu t \right)^{4}}} \right]} \boldsymbol{e}_{x}
$$
C. $$
\boldsymbol{J}=\frac{\varepsilon_{0} \nu}{\left( x_{0}-\nu t \right)} \boldsymbol{e}_{x}
$$
D. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{1} {\left( x_{0}-\nu t \right)^{2}}} \right]} \boldsymbol{e}_{x}
$$
E. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu} {\left( x_{0}+\nu t \right)^{3}}} \right]} \boldsymbol{e}_{x}
$$
F. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu} {\left( x_{0}+\nu t \right)^{2}}} \right]} \boldsymbol{e}_{x}
$$
G. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu^{2}} {\left( x_{0}-\nu t \right)}} \right]} \boldsymbol{e}_{x}
$$
H. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu^{2}} {\left( x_{0}-\nu t \right)^{2}}} \right]} \boldsymbol{e}_{x}
$$
I. $$
\boldsymbol{J}=\varepsilon_{0} U {\left[ {\frac{\nu} {\left( x_{0}-\nu t \right)^{4}}} \right]} \boldsymbol{e}_{x}
$$
J. $$
\boldsymbol{J}=\varepsilon_{0} \nu {\left[ {\frac{U} {\left( x_{0}-\nu t \right)}} \right]} \boldsymbol{e}_{x}
$$
Answer: A
The plates of a parallel-plate capacitor are circular, with a plate area of $S$ and a distance between the plates of $d$. The dielectric constant of the medium is $\varepsilon$, and the conductivity is $\gamma$. When the voltage between the plates is a DC voltage $U$, what is the Poynting vector at any point within the capacitor?

Options:
A. $$
-{\frac{\gamma U} {d}} e_{r}
$$
B. $$
-{\frac{\varepsilon \gamma U^{2}} {d}} e_{\theta}
$$
C. $$
-{\frac{\gamma U^{2}} {2 d^{2}}} e_{r}
$$
D. $$
{\frac{\varepsilon U^{2}} {2 d^{2}}} e_{r}
$$
E. $$
{\frac{\gamma U^{2}} {d}} e_{\theta}
$$
F. $$
{\frac{\gamma U} {2 d}} e_{\phi}
$$
G. $$
-{\frac{\gamma U^{3}} {2 d^{3}}} e_{r}
$$
H. $$
-{\frac{\varepsilon \gamma U} {2 d^{2}}} e_{r}
$$
I. $$
{\frac{\varepsilon U^{2}} {d^{2}}} e_{\phi}
$$
J. $$
{\frac{\gamma U^{2}} {d}} e_{\phi}
$$
Answer: C
Let a uniform plane wave propagate in a good conductor, where its propagation speed is 0.1% of the speed of light in a vacuum, and the wavelength is 0.3 mm. Given that the permeability of the medium is $\mu_{0}$, what are the frequency of the plane wave and the conductivity of the good conductor?

Options:
A. $$
3 . 5 \times1 0^{9} \, \mathrm{H z}
$$
$$
0 . 9 9 \times1 0^{7} \mathrm{S / m}
$$
B. $$
7 . 5 \times1 0^{8} \, \mathrm{H z}
$$
$$
1. 1 1 \times1 0^{6} \mathrm{S / m}
$$
C. $$
2 \times1 0^{9} \, \mathrm{H z}
$$
$$
9 . 5 0 \times1 0^{6} \mathrm{S / m}
$$
D. $$
1 . 5 \times1 0^{9} \, \mathrm{H z}
$$
$$
1. 3 0 \times1 0^{7} \mathrm{S / m}
$$
E. $$
5 \times1 0^{9} \, \mathrm{H z}
$$
$$
2 . 0 0 \times1 0^{7} \mathrm{S / m}
$$
F. $$
6 . 0 \times1 0^{9} \, \mathrm{H z}
$$
$$
1. 7 8 \times1 0^{7} \mathrm{S / m}
$$
G. $$
4 . 0 \times1 0^{8} \, \mathrm{H z}
$$
$$
2 . 5 0 \times1 0^{6} \mathrm{S / m}
$$
H. $$
1 \times1 0^{9} \, \mathrm{H z}
$$
$$
1. 1 1 \times1 0^{7} \mathrm{S / m}
$$
I. $$
2 . 5 \times1 0^{8} \, \mathrm{H z}
$$
$$
1. 0 0 \times1 0^{6} \mathrm{S / m}
$$
J. $$
1 \times1 0^{8} \, \mathrm{H z}
$$
$$
1. 0 5 \times1 0^{6} \mathrm{S / m}
$$
Answer: H
The electromagnetic parameters of medium 1 are given as $\varepsilon_{1}=5 \varepsilon_{0}, \mu_{1}=3 \mu_{0}, \sigma_{1}=0$, and medium 2 is considered an ideal conductor $(\sigma_{2}=\infty)$. Assume $y=0$ represents the surface of the ideal conductor, and the region $y > 0$ corresponds to medium 1. The electric field intensity in medium 1 is:
$$
E = e_{y} \, 20 \cos \left( 2 \times 10^{8} \, t - 2.58 \, z \right) \, \, \, \mathrm{V/m}.
$$
Calculate the value of $H$ at point $P$ when $t=6$ ns.

Options:
A. $$
=-\ e_{y} 6 1. 0 \times 1 0^{-3} \mathrm{A / m}
$$
B. $$
=-\ e_{x} 4 2.3 \times 1 0^{-3} \mathrm{A / m}
$$
C. $$
=-\ e_{z} 6 2. 3 \times 1 0^{-4} \mathrm{A / m}
$$
D. $$
=-\ e_{z} 6 2. 3 \times 2 0^{-3} \mathrm{A / m}
$$
E. $$
=-\ e_{x} 6 2. 3 \times1 0^{-3} \mathrm{A / m}
$$
F. $$
=-\ e_{x} 6 1. 5 \times 1 0^{-3} \mathrm{A / m}
$$
G. $$
=-\ e_{x} 7 2.3 \times 1 0^{-3} \mathrm{A / m}
$$
H. $$
=-\ e_{y} 6 2.3 \times 1 0^{-3} \mathrm{A / m}
$$
I. $$
=-\ e_{x} 6 2. 3 \times 1 0^{-2} \mathrm{A / m}
$$
J. $$
=-\ e_{x} 5 2. 3 \times 1 0^{-3} \mathrm{A / m}
$$
Answer: E
The complex vector of the electric field intensity of a uniform plane wave propagating in free space is given by
$$
E\ =\ e_{x} 10^{-4} \, \mathrm{e}^{-j 20 \pi z} \,+\, e_{y} 10^{-4} \, \mathrm{e}^{-j \left( 20 \pi z - \frac{\pi}{2} \right)} \, \mathrm{V/m}.
$$
Find: The average power passing through a unit area along the propagation direction is ().

Options:
A. $$
= 2. 6 5 \times 1 0^{-1 2} \ \mathbb{W} / \mathbb{m}^{2}
$$
B. $$
= 1. 9 7 \times 1 0^{-1 1} \ \mathbb{W} / \mathbb{m}^{2}
$$
C. $$
= 2. 6 5 \times 1 0^{-1 0} \ \mathbb{W} / \mathbb{m}^{2}
$$
D. $$
= 2. 7 3 \times 1 0^{-1 0} \ \mathbb{W} / \mathbb{m}^{2}
$$
E. $$
= 3. 0 5 \times 1 0^{-1 1} \ \mathbb{W} / \mathbb{m}^{2}
$$
F. $$
= 3. 1 4 \times 1 0^{-1 0} \ \mathbb{W} / \mathbb{m}^{2}
$$
G. $$
= 2. 4 8 \times 1 0^{-1 2} \ \mathbb{W} / \mathbb{m}^{2}
$$
H. $$
= 2. 8 4 \times 1 0^{-1 1} \ \mathbb{W} / \mathbb{m}^{2}
$$
I. $$
= 2. 6 5 \times1 0^{-1 1} \ \mathbb{W} / \mathbb{m}^{2}
$$
J. $$
= 2. 1 9 \times 1 0^{-1 2} \ \mathbb{W} / \mathbb{m}^{2}
$$
Answer: I
A uniform plane wave with a frequency of 150 MHz propagates in a lossy medium. It is given that $\varepsilon_{\mathrm{r}} = 1.4$, $\mu = 1$, and $\frac{\sigma}{\omega\varepsilon} = 10^{-4}$. After traveling a certain distance in the medium, how many meters does it take for the phase of the wave to change by 90°?

Options:
A. $$
= 0. 3 1 6 {\mathrm{~ m }}
$$
B. $$
= 0. 9 4 0 {\mathrm{~ m }}
$$
C. $$
= 0. 7 3 8 {\mathrm{~ m }}
$$
D. $$
= 0. 1 7 9 {\mathrm{~ m }}
$$
E. $$
= 0. 4 5 0 {\mathrm{~ m }}
$$
F. $$
= 0. 4 2 4 {\mathrm{~ m }}
$$
G. $$
= 0. 8 0 2 {\mathrm{~ m }}
$$
H. $$
= 0. 2 9 1 {\mathrm{~ m }}
$$
I. $$
= 0. 5 6 7 {\mathrm{~ m }}
$$
J. $$
= 0. 6 8 5 {\mathrm{~ m }}
$$
Answer: F
A uniform plane wave is incident perpendicularly from air onto the surface of a certain dielectric material. The standing wave ratio in the air is 2.7, and the surface of the dielectric corresponds to the point of minimum electric field in the standing wave. Determine the permittivity of the dielectric material.

Options:
A. $$
= 6. 2 1 \varepsilon_{0}
$$
B. $$
= 7. 5 4 \varepsilon_{0}
$$
C. $$
= 8. 7 6 \varepsilon_{0}
$$
D. $$
= 7. 2 7 \varepsilon_{0}
$$
E. $$
= 6. 4 9 \varepsilon_{0}
$$
F. $$
= 7. 0 5 \varepsilon_{0}
$$
G. $$
= 8. 1 8 \varepsilon_{0}
$$
H. $$
= 5. 3 6 \varepsilon_{0}
$$
I. $$
= 5. 8 9 \varepsilon_{0}
$$
J. $$
= 9. 0 3 \varepsilon_{0}
$$
Answer: D
A vertically polarized uniform plane wave is incident from underwater at an incident angle $\theta_{\mathrm{i}}=20^\circ$ onto the water-air interface. Given that fresh water has $\varepsilon_{\mathrm{t}}=81$, $\mu_{\mathrm{r}}=1$, and $\sigma=0$, determine the attenuation (in dB) of the transmitted wave after propagating one wavelength in the air.

Options:
A. $$
=-1 6 5. 4 \mathrm{~ d B}
$$
B. $$
=-1 7 1. 2 \mathrm{~ d B}
$$
C. $$
=-1 2 4. 5 \mathrm{~ d B}
$$
D. $$
=-1 7 9. 5 \mathrm{~ d B}
$$
E. $$
=-1 8 0. 1 \mathrm{~ d B}
$$
F. $$
=-1 5 8. 8 \mathrm{~ d B}
$$
G. $$
=-1 6 2. 3 \mathrm{~ d B}
$$
H. $$
=-1 3 9. 7 \mathrm{~ d B}
$$
I. $$
=-1 4 4. 8 \mathrm{~ d B}
$$
J. $$
=-1 5 0. 9 \mathrm{~ d B}
$$
Answer: F
The outer conductor radius of the coaxial line is $b=23 \ \mathrm{mm}$, the inner conductor radius is $a=10 \ \mathrm{mm}$, and the filling mediums are air and a lossless medium with $\varepsilon_{\mathrm{r}}=2.25$, respectively. Calculate its characteristic impedance.

Options:
A. $$
= 4 0. 0 5 \ \Omega
$$
B. $$
= 2 9. 8 5 \ \Omega
$$
C. $$
= 4 2. 1 8 \ \Omega
$$
D. $$
= 3 4. 6 7 \ \Omega
$$
E. $$
= 3 3. 3 2 \ \Omega
$$
F. $$
= 3 2. 9 1 \ \Omega
$$
G. $$
= 3 5. 4 4 \ \Omega
$$
H. $$
= 2 7. 7 2 \ \Omega
$$
I. $$
= 3 7. 0 6 \ \Omega
$$
J. $$
= 3 8. 3 3 \ \Omega
$$
Answer: E
It is known that the radiation power of a certain antenna is 100 $\mathbf{w}$, and the directivity factor is 3. Find the amplitude of the electric field strength in the maximum radiation direction at $r = 10$ km is ().

Options:
A. $$
17.50 \times 10^{-3}V/m
$$
B. $$
11.75 \times 10^{-3}V/m
$$
C. $$
16.25 \times 10^{-3}V/m
$$
D. $$
9.85 \times 10^{-3}V/m
$$
E. $$
18.00 \times 10^{-3}V/m
$$
F. $$
12.50 \times 10^{-3}V/m
$$
G. $$
15.10 \times 10^{-3}V/m
$$
H. $$
14.00 \times 10^{-3}V/m
$$
I. $$
10.90 \times 10^{-3}V/m
$$
J. $$
13.42 \times 10^{-3}V/m
$$
Answer: J
The magnetic field intensity $H$ of a uniform plane wave has an amplitude of ${\frac{1}{3 \pi}} \mathrm{A/m}$ and propagates in the direction $(-e_{1})$ in air with a phase constant of 30 rad/m. When $t=0$ and $z=0$, if $H$ is oriented as $(-e_{y})$, the expressions for $H$ is ().

Options:
A. $$
= e_{x} 4 0 \operatorname{c o s} ( 9 \times1 0^{9} t+3 0 z ) \mathrm{V / m}
$$
B. $$
= e_{y} 4 0 \operatorname{s i n} ( 9 \times10^{8} t+3 0 z ) \mathrm{V / m}
$$
C. $$
= e_{x} 4 0 \operatorname{s i n} ( 9 \times10^{9} t+3 5 z ) \mathrm{V / m}
$$
D. $$
= e_{z} 6 0 \operatorname{c o s} ( 7 \times10^{9} t+4 0 z ) \mathrm{V / m}
$$
E. $$
= e_{x} 5 0 \operatorname{c o s} ( 8 \times10^{9} t+2 0 z ) \mathrm{V / m}
$$
F. $$
= e_{x} 3 0 \operatorname{c o s} ( 1 \times10^{8} t+3 5 z ) \mathrm{V / m}
$$
G. $$
= e_{z} 3 0 \operatorname{c o s} ( 7 \times10^{8} t+3 5 z ) \mathrm{V / m}
$$
H. $$
= e_{y} 4 0 \operatorname{c o s} ( 9 \times10^{8} t+3 0 z ) \mathrm{V / m}
$$
I. $$
= e_{z} 5 0 \operatorname{c o s} ( 8 \times10^{9} t+2 5 z ) \mathrm{V / m}
$$
J. $$
= e_{y} 6 0 \operatorname{c o s} ( 2 \times10^{9} t+4 0 z ) \mathrm{V / m}
$$
Answer: A
The reaction between acetic acid and butanol produces butyl acetate, and the reaction equation is:
$(A){CH}_{3}COOH$ $+ (B)\mathrm{C}_{4}\mathrm{H}_{9}OH$ $\longrightarrow (P)\mathrm{CH}_{3}\mathrm{COOC}_{4}\mathrm{H}_{9} + (S)\mathrm{H}_{2}O$
When the reaction temperature is 100°C and sulfuric acid is used as a catalyst, the kinetics equation is:
$(-r_{\Lambda}) = kc_{\Lambda}^{2}$
At this time, the reaction rate constant $k = 17.4 \times 10^{-3}\,\mathbb{L}/($mol$\cdot$min$)$.
It is known that the process takes place in an ideal batch reactor. If the initial concentrations of acetic acid in the feed are 0.9 mol/L and 1.8 mol/L respectively, and the volume of the reaction mixture is 100 L, how many kilograms of butyl acetate are obtained in each case?

Options:
A. 5.15kg,10.30kg
B. 5.10kg,10.20kg
C. 4.80kg,9.60kg
D. 4.90kg,9.80kg
E. 5.25kg,10.50kg
F. 5.40kg,10.80kg
G. 6.00kg,12.00kg
H. 4.70kg,9.40kg
I. 5.22kg,10.44kg
J. 5.00kg,10.00kg
Answer: I
A one-step synthesis reaction takes place in a Plug Flow Reactor (PFR). Given the reaction
$$
\mathrm{A + B} \longrightarrow \mathrm{P}
$$
occurs at 450°C and atmospheric pressure, with $k = 0.1524 \; \mathrm{L / (mol \cdot s)}$.
If the reactants are fed into the reactor at an equimolar ratio, calculate the space-time $\tau$ and residence time $t$ required for a conversion rate of 10%.

Options:
A. $$
\tau=70.00s
t=84.20s
$$
B. $$
\tau=82.00s
t=88.00s
$$
C. $$
\tau=81.00s
t=85.20s
$$
D. $$
\tau=82.04s
t=90.00s
$$
E. $$
\tau=78.50s
t=87.40s
$$
F. $$
\tau=83.30s
t=82.10s
$$
G. $$
\tau=75.30s
t=80.50s
$$
H. $$
\tau=82.04s
t=84.20s
$$
I. $$
\tau=90.10s
t=85.40s
$$
J. $$
\tau=85.00s
t=81.00s
$$
Answer: H
A certain liquid-phase reaction takes place in a stirred tank with a volume of $V_{\mathrm{R}}$. The reaction is a first-order irreversible reaction with a rate constant of $k$ and a feed flow rate of $v$. The conversion rate of the reactant is calculated as $x_{\mathrm{A}}$.
If half of the outlet material is recycled, and the volume of the fresh feed is correspondingly reduced by half, what is the production rate of the output stream?

Options:
A. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{kc_{A0}V_{\mathrm{R}}}{1+2k\frac{V_{\mathrm{R}}}v}
$$
B. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{2kc_{A0}V_{\mathrm{R}}}{1+k\frac{V_{\mathrm{R}}}v}
$$
C. $$
c_{A0}x_{A\mathrm{f}}\frac v4=\frac{kc_{A0}V_{\mathrm{R}}}{1+2k\frac{V_{\mathrm{R}}}v}
$$
D. $$
c_{A0}x_{A\mathrm{f}}v=\frac{c_{A0}V_{\mathrm{R}}}{1+k\frac{V_{\mathrm{R}}}v}
$$
E. $$
c_{A0}x_{A\mathrm{f}}v=\frac{kV_{\mathrm{R}}}{1+2\frac{V_{\mathrm{R}}}v}
$$
F. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{c_{A0}V_{\mathrm{R}}}{1+2\frac{kV_{\mathrm{R}}}v}
$$
G. $$
c_{A0}x_{A\mathrm{f}}v=\frac{2kc_{A0}V_{\mathrm{R}}}{1+k\frac{V_{\mathrm{R}}}v}
$$
H. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{k^2c_{A0}V_{\mathrm{R}}^2}{1+2k\frac{V_{\mathrm{R}}}{v}}
$$
I. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{kc_{A0}V_{\mathrm{R}}}{1+k\frac{V_{\mathrm{R}}}{v}}
$$
J. $$
c_{A0}x_{A\mathrm{f}}\frac v2=\frac{kc_{A0}V_{\mathrm{R}}}{1+4k\frac{V_{\mathrm{R}}}v}
$$
Answer: A