CP Two capacitor Figure P26.69 plates with area A are separated by a distance d. The spac

step 1 For this problem, we have an RC circuit with a discharging capacitor. So for part A, we want the

CP Two capacitor Figure P26.69 plates with area A are...

CP Two capacitor } & \text { Figure P26.69 }\end{array}$ plates with area $A$ are separated by a distance $d$. The space between the plates is filled with dielectric material with dielectric constant $K$ and resistivity $\rho$. This capacitor is attached to $\quad$ Dielectric a battery that supplies a con- $\quad$ constant $K,$ stant potential $\mathcal{E},$ as shown $\quad$ resistivity $\hat{t}$ in Fig. P26.69, and is fully charged. At time $t=0$ the switch $S$ is opened. Owing to the nonzero value of $\rho,$ this capacitor discharges by leaking. We can model this device as a capacitor and a resistor in parallel. (a) Analyze this circuit and determine the time constant, characterizing the discharge in terms of the parameters given above. Now assume the capacitor has plates with area $A=1.00 \mathrm{~cm}^{2}$ separated by $d=30.0 \mu \mathrm{m}$ and is filled with a ceramic of dielectric constant $K=8.70$ and resistivity $\rho=3.10 \times 10^{12} \Omega \cdot \mathrm{m}$. (b) If $\mathcal{E}=5.00 \mathrm{~V},$ what is the charge on the capacitor at $t=0 ?$ (c) At what time will the capacitor have half of its original charge? (d) What is the magnitude of the leaking current at that point?

CP Two capacitor } & \text { Figure P26.69 }\end{array}$ plates with area $A$ are separated by a distance $d$. The space between the plates is filled with dielectric material with dielectric constant $K$ and resistivity $\rho$. This capacitor is attached to $\quad$ Dielectric a battery that supplies a con- $\quad$ constant $K,$ stant potential $\mathcal{E},$ as shown $\quad$ resistivity $\hat{t}$ in Fig. P26.69, and is fully charged. At time $t=0$ the switch $S$ is opened. Owing to the nonzero value of $\rho,$ this capacitor discharges by leaking. We can model this device as a capacitor and a resistor in parallel. (a) Analyze this circuit and determine the time constant, characterizing the discharge in terms of the parameters given above. Now assume the capacitor has plates with area $A=1.00 \mathrm{~cm}^{2}$ separated by $d=30.0 \mu \mathrm{m}$ and is filled with a ceramic of dielectric constant $K=8.70$ and resistivity $\rho=3.10 \times 10^{12} \Omega \cdot \mathrm{m}$.
(b) If $\mathcal{E}=5.00 \mathrm{~V},$ what is the charge on the capacitor at $t=0 ?$ (c) At what time will the capacitor have half of its original charge? (d) What is the magnitude of the leaking current at that point?

Key Concepts

RC Circuit Analysis

An RC circuit combines resistive and capacitive elements to analyze how charge and voltage change over time. When leakage is present, the capacitor and the resistive path are considered in parallel, leading to discharge behavior that is governed by the interplay between the capacitance and the effective resistance.

Exponential Decay

Exponential decay describes how the charge and voltage across a capacitor decrease over time in an RC circuit. The mathematical model, typically expressed as Q(t) = Q? e^(?t/?), shows that the capacitor loses charge at a rate proportional to its remaining charge, leading to a predictable, continuous decay pattern.

Time Constant

The time constant in an RC circuit is a key parameter defined by the product of the resistance and the capacitance (? = RC). It characterizes the rate at which the capacitor discharges, with larger time constants indicating slower discharge rates. The time constant determines the exponential time scale over which the charge, voltage, and current decay.

Resistivity and Dielectric Leakage

Resistivity is a measure of how strongly a material opposes the flow of electric current. In the context of capacitors, even insulators (dielectrics) have a finite resistivity, which can allow a small leakage current that causes the capacitor to slowly discharge. This leakage is modeled by treating the dielectric as providing a resistive path between the capacitor plates.

Dielectrics

A dielectric is an insulating material placed between the plates of a capacitor that increases the capacitor’s ability to store charge. The effectiveness of a dielectric is quantified by its dielectric constant, which scales the capacitance relative to a vacuum. Dielectrics also influence leakage characteristics due to their inherent resistivity.

Capacitors

A capacitor is an electrical component that stores energy in an electric field between two conducting plates separated by an insulating material. Its ability to store charge is measured by its capacitance, which depends on the area of the plates, the separation distance, and the properties of the dielectric material between them.

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