電容：練習題

Which of the following is the correct unit for the time constant of an \( RC \) circuit?

A \( 100 \mu\text{F} \) capacitor is discharged through a \( 47 \text{ k}\Omega \) resistor. Calculate the time taken for the potential difference across the capacitor to fall to \( e^{-1} \) (approximately \( 37\% \)) of its initial value.

A capacitor of capacitance \(C\) is charged to a potential difference \(V\). It is then discharged through a resistor of resistance \(R\). After a time \(t = RC \ln(2)\), what fraction of the initial energy stored in the capacitor remains?

A parallel-plate capacitor has a capacitance of \( 2.5 \text{ nF} \). It is connected to a \( 12 \text{ V} \) DC power supply. Calculate the charge stored on one of the plates of the capacitor.

A parallel-plate capacitor with plate area \(A\) and separation \(d\) is charged to a potential difference \(V\) and then isolated. A dielectric slab of relative permittivity \(\varepsilon_r = 3.0\) is then inserted to completely fill the space between the plates. What is the ratio of the final energy stored in the capacitor to the initial energy stored?

A capacitor of capacitance \( C \) is charged to a potential difference \( V \) and then discharged through a resistor \( R \). Derive an expression for the time taken for the charge to fall to \( \frac{1}{e^2} \) of its initial value in terms of the time constant.

A student uses a power supply to charge a capacitor of capacitance \( C \) to a potential difference \( V \). The energy stored in the capacitor is \( E \).

(a) State the relationship between the charge \( Q \) stored on the plates, the capacitance \( C \), and the potential difference \( V \).

(b) The potential difference across the capacitor is now doubled to \( 2V \). Show that the new energy stored in the capacitor is \( 4E \).

(c) A second, identical capacitor is connected in parallel with the first capacitor (which is still charged to \( 2V \)). Calculate the total capacitance of this combination in terms of \( C \).

A technician is evaluating the suitability of a capacitor for a specific timing circuit. A \( 470 \mu\text{F} \) capacitor is fully charged to a potential difference of \( 9.0 \text{ V} \) and then discharged through a \( 150 \text{ k}\Omega \) resistor.

(a) Calculate the time constant of the discharge circuit and the initial energy stored in the capacitor.

(b) Determine the time taken for the potential difference across the capacitor to fall to \( 2.0 \text{ V} \).

(c) Calculate the current in the circuit at the instant the potential difference is \( 2.0 \text{ V} \).