Quiz: Capacitors, Limits, and Reproduction — 23 questions

1. Regarding capacitor charging fundamentals, which statements are correct?

A. A resistor carrying constant current has voltage $u_R=\frac{I}{R}$.

B. A capacitor charged by constant current has charge $q=It$.

C. For zero initial voltage, constant-current charging gives $u_C(t)=\frac{I}{C}t$.

D. A capacitor charged by constant current has exponentially increasing voltage.

E. During constant-current charging, capacitor voltage remains constant while resistor voltage increases.

Answer: (B), (C)

Explanation

With constant-current charging, charge increases as $q=It$ and voltage increases linearly as $u_C(t)=\frac{I}{C}t$. The resistor voltage under constant current is constant, whereas the capacitor voltage changes with time; resistor-capacitor charging is exponential rather than linear.

2. A constant current passes through a resistor and charges a capacitor; which statements are correct?

A. The capacitor voltage is described by an exponential law under constant current.

B. The capacitor voltage remains fixed while the resistor voltage changes with time.

C. The capacitor voltage increases linearly with time during constant-current charging.

D. The resistor voltage increases linearly as the capacitor charges.

E. The resistor voltage is determined by $u_R=RI$.

Answer: (C), (E)

Explanation

The resistor relation is $u_R=RI$, so its voltage is constant when both resistance and current are constant. In contrast, constant-current capacitor charging gives a voltage that increases linearly with time.

3. Which statements correctly distinguish ideal current and voltage generators?

A. A real cell maintains a terminal voltage equal to $E$ for every current.

B. An ideal voltage generator maintains a constant potential difference between terminals.

C. An ideal voltage generator is described by $u_{PN}=E-rI$.

D. The ammeter reading of an ideal current generator remains constant.

E. An ideal current generator delivers a constant current intensity.

Answer: (B), (D), (E)

Explanation

An ideal current generator maintains a constant current, so its ammeter reading does not change. An ideal voltage generator maintains $u_{PN}=E$, while a real cell follows $u_{PN}=E-rI$ and includes an internal voltage drop.

4. A real cell supplies a current to an external circuit; which statements are correct?

A. The internal voltage drop is represented by $E-r$.

B. The term $rI$ represents the internal voltage drop.

C. Its terminal voltage is modeled by $u_{PN}=E-rI$.

D. Its terminal voltage remains equal to $E$ as current changes.

E. The electromotive force is represented by $rI$.

Answer: (B), (C)

Explanation

A real cell is modeled by $u_{PN}=E-rI$, where $E$ is the electromotive force and $rI$ is the internal voltage drop. The relation is therefore not the ideal fixed-voltage model $u_{PN}=E$.

5. For a charging circuit containing resistances $R$ and $R_1$, which statements are correct?

A. The total resistance in the differential model is $R+R_1$.

B. The charge satisfies $\frac{dq}{dt}+\frac{q}{(R+R_1)C}=\frac{E}{R+R_1}$.

C. The differential equation uses $R-R_1$ as the circuit resistance.

D. The charge equation contains the coefficient $\frac{1}{(R+R_1)C}$.

E. The source term in the equation is $\frac{E}{R+R_1}$.

Answer: (A), (B), (D), (E)

Explanation

Applying Kirchhoff’s law to the charging circuit gives $\frac{dq}{dt}+\frac{q}{(R+R_1)C}=\frac{E}{R+R_1}$. The total resistance is the sum $R+R_1$, and the source term is divided by that resistance.

6. Regarding the constants in an RC charging model, which statements are correct?

A. The forcing constant is $h=\frac{E}{R+R_1}$.

B. The time constant uses the total resistance multiplied by capacitance.

C. The forcing constant depends on capacitance and total resistance.

D. The time constant is $\tau=(R+R_1)C$.

E. The time constant depends on electromotive force and resistance.

Answer: (A), (B), (D)

Explanation

The time constant is $\tau=(R+R_1)C$, so it depends on the resistance and capacitance. The forcing constant is $h=\frac{E}{R+R_1}$, so it depends on the source electromotive force and resistance.

7. For an initially uncharged capacitor in an RC charging circuit, which statements are correct?

A. The charge decreases exponentially toward zero during charging.

B. The voltage across $R_1$ is $u_{R1}(t)=R_1h e^{-t/\tau}$.

C. The charge is $q(t)=\tau h\left(1-e^{-t/\tau}\right)$.

D. The voltage across $R_1$ approaches the final value $\tau h$.

E. The charge approaches the final value $\tau h$ as time increases.

Answer: (B), (C), (E)

Explanation

For an initially uncharged capacitor, the charge follows $q(t)=\tau h\left(1-e^{-t/\tau}\right)$ and approaches the final value $\tau h$. The current decays exponentially toward zero, whereas the charge does not decay; the voltage across $R_1$ is given by $u_{R1}(t)=R_1h e^{-t/\tau}$.

8. Regarding experimental identification of an RC circuit, which proposition is correct?

A. The 99.9% charging duration is approximately $\Delta t=7(R+R_1)C$.

B. The 63% charging level is reached after approximately $0.1\tau$.

C. The 63% charging level is reached after approximately $7\tau$.

D. The 99.9% charging duration is approximately $\Delta t=(R+R_1)/C$.

E. The 99.9% charging duration is approximately $\Delta t=\tau=(R+R_1)C$.

Answer: (A)

Explanation

At 99.9% charge, the duration is approximately seven time constants, so $\Delta t=7(R+R_1)C$. The 63% charging level corresponds instead to approximately one time constant.

9. Concerning the parameters obtained from the experimental straight line, select the exact propositions:

A. The fixed resistance obtained from the straight line is $20\,\Omega$.

B. The capacitance obtained from the straight line is $20\,\mu\mathrm{F}$.

C. The capacitance obtained from the straight line is $100\,\mu\mathrm{F}$.

D. The slope determines the fixed resistance, while the intercept determines capacitance.

E. The fixed resistance obtained from the straight line is $100\,\Omega$.

Answer: (B), (E)

Explanation

The experimental straight line gives $C=20\,\mu\mathrm{F}$ and $R_1=100\,\Omega$. The slope determines the capacitance, while the intercept determines the fixed resistance.

10. The electric energy stored in a capacitor is described by which exact propositions?

A. It can be written as $E_e=\frac12C^2u_C^2$.

B. It can be written as $E_e=\frac{q^2}{C}$.

C. It can be written as $E_e=\frac{q^2}{2C}$.

D. It can be written as $E_e=\frac12Cu_C^2$.

E. The capacitor current directly gives the stored energy without charge or voltage.

Answer: (C), (D)

Explanation

The stored energy is $E_e=\frac{q^2}{2C}=\frac12Cu_C^2$. Current controls how charge and voltage change, rather than directly giving the stored-energy expression.

11. A constant current charges an initially uncharged capacitor. Which propositions are exact?

A. The stored energy increases quadratically with time.

B. The capacitor voltage increases quadratically with time.

C. The stored energy is $E_e=\frac{I^2}{2C}t^2$.

D. The stored energy increases linearly with time.

E. The stored energy is $E_e=\frac{I}{2C}t^2$.

Answer: (A), (C)

Explanation

With constant current and an initially uncharged capacitor, energy follows $E_e=\frac{I^2}{2C}t^2$. Thus energy varies quadratically with time, whereas voltage varies linearly.

12. For a parallel-plate capacitor, which propositions concerning capacitance and permittivity are exact?

A. The relative permittivity is $\varepsilon_r=\frac{\varepsilon_0}{\varepsilon}$.

B. The capacitance is $C=\varepsilon\frac{S}{e}$.

C. The capacitance is independent of the plate area $S$.

D. The capacitance is $C=\varepsilon\frac{e}{S}$.

E. The relative permittivity is $\varepsilon_r=\frac{\varepsilon}{\varepsilon_0}$.

Answer: (B), (E)

Explanation

For parallel plates, capacitance is $C=\varepsilon\frac{S}{e}$, and relative permittivity is $\varepsilon_r=\frac{\varepsilon}{\varepsilon_0}$. These expressions distinguish absolute permittivity from relative permittivity.

13. For a capacitor charged by a constant current, which proposition is correct?

A. The two plates carry unequal charge magnitudes.

B. The charge magnitude satisfies $q=It$.

C. Both plates carry charge $+q$.

D. The charge magnitude satisfies $q=I/t$.

E. One plate carries $+q$ and the other carries $-q$.

Answer: (B)

Explanation

Constant-current charging gives $q=It$. The two plates carry equal charge magnitudes with opposite signs: one has $+q$ and the other has $-q$.

14. Regarding the dielectric relation for the capacitor model, select the exact propositions:

A. The capacitance is $C=\varepsilon\frac{S}{e}$.

B. The capacitance is $C=\varepsilon\frac{e}{S}$.

C. The capacitance is independent of dielectric thickness.

D. Increasing dielectric thickness increases the capacitance.

E. Increasing plate area increases the capacitance.

Answer: (A), (E)

Explanation

The dielectric model gives $C=\varepsilon\frac{S}{e}$. Increasing plate area increases capacitance, whereas increasing dielectric thickness decreases it.

15. Regarding limits and continuity methods, select the exact statements:

A. The squeeze theorem requires two bounding functions with different limits.

B. Continuity at a point requires the limit to equal the function value.

C. A limit describes function behavior near a point or toward infinity.

D. A limit exists at a point whenever the function value is defined there.

E. A limit problem may involve establishing a bound or an equivalent expression.

Answer: (B), (C), (E)

Explanation

The squeeze theorem applies when the target function lies between two functions with the same limit. A limit concerns nearby behavior, whereas continuity additionally requires agreement with the function value.

16. Which statements correctly describe methods for evaluating limits?

A. Interpreting a limit may provide meaning after its value is determined.

B. A finite function value guarantees that the corresponding limit exists.

C. An equivalent expression can simplify the determination of a limit.

D. The squeeze theorem applies when bounding functions share the same limit.

E. Continuity follows from limit existence without checking the function value.

Answer: (A), (C), (D)

Explanation

A limit problem can be approached by finding a bound or an equivalent expression and then determining the resulting limit. The squeeze theorem needs equal limiting values for both bounds, and continuity requires equality with the function value.

17. A function is continuous at a point when:

A. Its limit may fail to exist provided the function value is finite.

B. Its limit exists, even if it differs from the function value.

C. Its limit at the point exists and equals its function value.

D. Its nearby values approach the value assigned at the point.

E. Its function value is defined, regardless of nearby behavior.

Answer: (C), (D)

Explanation

Continuity at a point means that the limit exists and equals the function value at that point. Merely defining the function value or having a finite limit does not establish continuity without this equality.

18. Concerning continuity, asymptotes, and variations, which statements are exact?

A. Continuity alone guarantees uniqueness of a solution on an interval.

B. A finite limit at infinity can identify a horizontal asymptote.

C. A divergent quotient such as $\frac{f(x)}{x}$ can identify a branch direction.

D. Continuity at a joining point involves both one-sided limits and the function value.

E. Strict monotonicity is used to guarantee uniqueness of a solution.

Answer: (B), (C), (D), (E)

Explanation

For a piecewise function, continuity at a joining point is tested by comparing the left-hand limit, right-hand limit, and function value. In existence-and-uniqueness arguments, continuity supports existence under endpoint conditions, while strict monotonicity supports uniqueness.

19. Which statements correctly express the geometric conclusions associated with the complex ratios?

A. The corresponding ratio is purely imaginary and indicates perpendicular directions.

B. The real ratio indicates that the relevant directions are collinear.

C. The area of rhombus OACD is $6\sin\theta$.

D. Triangle ABM is established as right-angled at M.

E. The ratio $\frac{z_M-z_B}{z_M-z_A}=\frac{1}{\sqrt{6}}$ is real.

Answer: (B), (D), (E)

Explanation

For the specified points, OACD is a rhombus with area $6\sin\theta$. The ratio $\frac{z_M-z_B}{z_M-z_A}=\frac{1}{\sqrt{6}}$ is real and is used to establish that triangle ABM is right-angled at M.

20. Regarding male reproductive regulation, which statements are correct?

A. Testosterone is secreted by the pituitary gland.

B. Gonadoliberin is secreted by hypothalamic neurons.

C. Spermatogenesis is supported by the hypothalamus.

D. Gonadoliberin acts on receptors located on pituitary cells.

E. Gonadostimulins are secreted by testicular cells.

Answer: (B), (D)

Explanation

Gonadoliberin, or GnRH, is released by the hypothalamus and acts on pituitary receptors. Gonadostimulins are pituitary hormones, whereas the testis produces testosterone and supports spermatogenesis.

21. After castration of a pubescent male animal, which statements are correct?

A. Castration causes hypertrophy of the accessory glands.

B. The corpus luteum develops from a testicular follicle after castration.

C. Accessory gland atrophy may follow removal of the testes.

D. Loss of testicular feedback contributes to gonadostimulin hypersecretion.

E. Castration can cause gonadostimulin hypersecretion.

Answer: (C), (D), (E)

Explanation

Removing the testes eliminates testicular negative feedback, causing gonadostimulin hypersecretion. The accessory glands subsequently undergo atrophy, and the corpus luteum is a post-ovulatory follicular structure rather than a result of castration.

22. In an experimental GnRH injection, which observations are correct?

A. S2 shows an LH increase from 0.51 to approximately 9.

B. S3 testosterone rises from 6.9 to approximately 16.

C. S2 testosterone decreases from 6 to approximately 3.

D. S3 LH remains close to 0.6 after injection.

E. S2 remains unchanged in both LH and testosterone.

Answer: (A), (D)

Explanation

S2 responds strongly to GnRH, with LH rising from 0.51 to 9 and testosterone from 6 to 16. S3 remains approximately unchanged, indicating no comparable response in that subject.

23. Which statements correctly apply experimental diagnosis of male infertility?

A. The normal testosterone reference range is 10–30 nmol/L.

B. A GnRH response suggests a deficient hypothalamic signal.

C. A GnRH response demonstrates an irreversible testicular failure.

D. Experimental diagnosis can compare germ-cell DNA content and cell numbers.

E. The normal LH reference range is 1–9 mU/L.

Answer: (A), (B), (D), (E)

Explanation

A response to discontinuous GnRH indicates deficient hypothalamic signaling with an intact pituitary-testis pathway. The stated reference ranges are LH 1–9 mU/L and testosterone 10–30 nmol/L, while experimental diagnosis also uses germ-cell analysis and hormone testing.