Physics

Chapter 8: Capacitance

Imron Rosyadi

Chapter 8: Capacitance

Table of Contents

• 8.1 Capacitors and Capacitance
  • Introduction to Capacitors
  • Calculating Capacitance
• 8.2 Capacitors in Series and in Parallel
  • Series Combinations
  • Parallel Combinations
• 8.3 Energy Stored in a Capacitor
  • Energy Storage Principles
  • Energy Density
• 8.4 Capacitor with a Dielectric
  • Effects of Dielectrics
  • Dielectric Constant
• 8.5 Molecular Model of a Dielectric
  • Polarization at Molecular Level
  • Dielectric Breakdown

Learning Objectives

By the end of this chapter, you will be able to:

• Explain the concepts of a capacitor and its capacitance.
• Describe how to evaluate the capacitance of conductor systems.
• Determine equivalent capacitance for series and parallel combinations.
• Compute potential difference, charge, and net capacitance in capacitor networks.
• Explain how energy is stored in a capacitor and use energy relations.
• Describe the effects of a dielectric on capacitance and other properties.
• Calculate capacitance of capacitors containing a dielectric.
• Explain polarization of a dielectric and its effect on electric fields.
• Explain dielectric breakdown.

This slide outlines the key learning objectives for Chapter 8 on Capacitance. Each bullet point represents a core concept or skill students should master by the end of the chapter. We will cover these in detail throughout the presentation.

8.1 Capacitors and Capacitance

What is a Capacitor?

• Definition: A device used to store electrical charge and electrical energy.
• Structure: Typically two electrical conductors (plates/electrodes) separated by a distance.
• Interspace:
  • Can be a vacuum (vacuum capacitor).
  • Usually filled with an insulating material called a dielectric.
• Storage Capacity: Determined by its capacitance.

Note

Capacitors are fundamental components in electronics, found everywhere from radio receivers to medical defibrillators.

Figure 8.2: (a) Parallel-plate capacitor. (b) Rolled capacitor.

A capacitor is a passive electronic component that stores energy in an electric field. Think of it as a temporary battery that can quickly release its stored energy. The basic structure involves two conductive plates separated by an insulator, which we call a dielectric. The amount of charge and energy a capacitor can store is quantified by its capacitance. These devices are ubiquitous in modern technology.

How Capacitors Work

• When connected to a battery:
  1. Battery potential moves a small amount of charge \(Q\) from the positive plate to the negative plate.
  2. Capacitor remains neutral overall.
  3. Plates acquire charges \(+Q\) and \(-Q\).
• Electric Field:
  • Between parallel plates, the electric field \(E\) is proportional to charge \(Q\).
  • \(E = \sigma / \varepsilon_0\), where \(\sigma\) is surface charge density (\(Q/A\)).
• Capacitance (C):
  • Ratio of maximum charge \(Q\) to applied voltage \(V\).
  • \(C = Q / V\) (Equation 8.1).
  • \(V\) represents the potential difference (\(\Delta V\)).

Figure 8.3: Electric field lines in a parallel-plate capacitor.

When a capacitor is connected to a voltage source like a battery, charge moves from one plate to the other, creating an electric field between them. The positive plate gets \(+Q\) charge and the negative plate gets \(-Q\). The strength of this electric field is directly related to the amount of charge stored. Capacitance is essentially a measure of how much charge a capacitor can hold for a given voltage across its plates. It’s a fundamental property of the capacitor’s design.

Units of Capacitance

• SI Unit: Farad (F), named after Michael Faraday.
  \(1\ \text{F} = 1\ \text{C}/1\ \text{V}\)

• Large Unit: 1 Farad is a very large capacitance.
  A 1.0‑F capacitor stores 1.0 C of charge at 1.0 V.

• Typical Values: Capacitance values usually range from:

  • Picofarads (pF): \(1\ \text{pF} = 10^{-12}\ \text{F}\)
  • Nanofarads (nF): \(1\ \text{nF} = 10^{-9}\ \text{F}\)
  • Microfarads (µF): \(1\ \text{µF} = 10^{-6}\ \text{F}\)
  • Millifarads (mF): \(1\ \text{mF} = 10^{-3}\ \text{F}\)

Figure 8.4: Typical capacitors used in electronic devices.

The Farad is a surprisingly large unit, which is why we usually work with prefixes like micro, nano, or pico Farads. This slide visually demonstrates the variety of shapes and sizes capacitors come in, highlighting that physical size isn’t always indicative of capacitance. This also reinforces the practical aspect of capacitors in everyday electronics.

Problem-Solving Strategy: Calculating Capacitance

1. Assume charge \(Q\): Start by assuming the capacitor has a charge \(Q\).
2. Determine electric field \(\mathbf{E}\): Calculate the electric field between the conductors. Gauss’s law can be useful for symmetric arrangements.
3. Find potential difference \(V\): Use the integral \(V = \left|\int_{A}^{B} \mathbf{E}\cdot d\mathbf{l}\right|\) where the path goes from one conductor to the other.
4. Calculate capacitance \(C\): Use the definition \(C = \frac{Q}{V}\).

Tip

Capacitance is a function only of the geometry and the material between the plates, not of Q or V.

This problem-solving strategy is a systematic approach to calculating capacitance for various conductor geometries. We’ll apply this strategy to parallel-plate, spherical, and cylindrical capacitors. The key idea is that capacitance is an intrinsic property based on the physical design, independent of the actual charge or voltage it’s currently experiencing.

Capacitance of a Parallel-Plate Capacitor

• Setup: Two identical conducting plates, area \(A\), separated by distance \(d\).
• Observations:
  • Larger \(A\) means more charge stored, so \(C\) increases with \(A\).
  • Smaller \(d\) means stronger attraction, so \(C\) increases with decreasing \(d\).
• Surface Charge Density: \(σ = \frac{Q}{A}\)
• Electric Field (uniform): \(E = \frac{σ}{\varepsilon_0}\)
  • \(\varepsilon_0 = 8.85 \times 10^{-12}\ \text{F/m}\) (permittivity of free space).
• Potential Difference: \(V = Ed = \frac{σ d}{\varepsilon_0} = \frac{Q d}{\varepsilon_0 A}\)
• Capacitance: \[C = \frac{Q}{V} = \frac{Q}{Qd / (ε₀A)} = \frac{ε₀A}{d}\] (Equation 8.3)

Figure 8.5: Parallel-plate capacitor geometry.

For a parallel-plate capacitor, the capacitance is directly proportional to the plate area and inversely proportional to the separation distance. This formula, derived from first principles using electric field and potential, is crucial for understanding these basic capacitors. It highlights how physical dimensions directly dictate the capacitance value.

Example 8.1: Parallel-Plate Capacitor Calculations

Problem:

1. What is the capacitance of an empty parallel‑plate capacitor with plates of area \(1.00\ \text{m}^2\) separated by \(1.00\ \text{mm}\)?
2. How much charge is stored if \(3.00\times10^{3}\ \text{V}\) is applied?

Solution: (a) Using \(C = \varepsilon_0 A/d\): \[C = (8.85 \times 10^{-12} \text{ F/m}) \frac{1.00 \text{ m}^2}{1.00 \times 10^{-3} \text{ m}} = 8.85 \times 10^{-9} \text{ F} = 8.85 \text{ nF}\]

2. Using \(Q = CV\): \[Q = (8.85 \times 10^{-9} \text{ F})(3.00 \times 10^3 \text{ V}) = 26.6 \times 10^{-6} \text{ C} = 26.6 \text{ μC}\]

Important

This example demonstrates that even with a large plate area, typical capacitance values are often in the nanofarad range, emphasizing that 1 F is a very large unit.

This example applies the formula for parallel-plate capacitance. Part (a) shows that even with a significant plate area, the capacitance is still relatively small, in the nanofarad range. Part (b) then uses this capacitance to calculate the charge stored for a given voltage. This helps students to get a feel for the magnitudes involved in practical capacitor applications.

Capacitance of a Spherical Capacitor

• Setup: Two concentric conducting spherical shells.
  • Inner radius \(R_1\), outer radius \(R_2\).
    
  • Charges \(+Q\) (inner) and \(-Q\) (outer).
    
• Electric Field (between shells): From Gauss’s Law:
  \[\vec{E} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]
• Potential Difference (from R₂ to R₁): \[V = -\int_{R_2}^{R_1} E⃗ ⋅ dl⃗ = \frac{Q}{4πε₀} \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\]
• Capacitance: \[C = \frac{Q}{V} = 4πε₀ \frac{R₁R₂}{R₂ - R₁}\] (Equation 8.4)

Figure 8.6: Spherical capacitor geometry.

Next, we apply the capacitance calculation strategy to a spherical capacitor. By using Gauss’s law for the electric field and then integrating to find the potential difference, we arrive at the capacitance formula. This illustrates how capacitance depends on the radii of the two concentric spheres, and it’s a good exercise in applying fundamental principles to a different geometry.

Capacitance of a Cylindrical Capacitor

• Setup: Two concentric conducting cylinders.
  • Inner radius \(R_1\), outer radius \(R_2\), length \(l\).
  • Charges \(+Q\) (inner) and \(-Q\) (outer).
• Electric Field (between cylinders): From Gauss’s Law: \[ \vec{E} = \frac{1}{2\pi\varepsilon_0} \frac{Q}{rl} \hat{r} \] (Equation 8.5)
• Potential Difference (from \(R_1\) to \(R_2\) ): \[ V = \int_{R_1}^{R_2} \mathbf{E} \cdot d\mathbf{l} = \frac{Q}{2\pi\varepsilon_0\,l} \ln\!\left(\frac{R_2}{R_1}\right) \]
• Capacitance: \[C = \frac{Q}{V} = \frac{2πε₀l}{\ln(R₂/R₁)}\] (Equation 8.6)

Figure 8.7: Cylindrical capacitor geometry.

The cylindrical capacitor is another common geometry, particularly relevant for coaxial cables. Again, we follow the same problem-solving strategy: find the electric field using Gauss’s law for cylindrical symmetry, then integrate to find the potential difference, and finally use the definition of capacitance. This result shows how capacitance depends on the length and the ratio of the radii.

Types of Commercial Capacitors

• Common Capacitors:
  • Two metal foils separated by insulation (e.g., mica, ceramic, paper, Teflon™).
  • Encased in a protective coating with leads for circuits.
• Electrolytic Capacitors:
  • Oxidized metal in a conducting paste.
  • High capacitance relative to size.
  • Polarized: Must be connected with correct polarity; reverse polarization destroys oxide film. Not suitable for AC circuits.

• Variable Air Capacitors:
  • Two sets of parallel plates: fixed (stator) and rotatable (rotor).
  • Capacitance tuned by changing overlap area.
  • Applications: Radio tuning.

Figure 8.8: Variable air capacitor.

Beyond the theoretical geometries, it’s important to understand practical capacitor types. This slide introduces common designs like film capacitors, specialized electrolytic capacitors with high capacitance but polarity sensitivity, and variable capacitors used for tuning circuits. The practical examples help bridge the gap between theoretical physics and real-world applications.

Circuit Representations

• General Capacitor:
  • Symbol in (a) is most common.
  • Resembles a parallel-plate capacitor.
• Electrolytic Capacitor:
  • Symbol in (b).
  • Curved plate indicates the negative terminal, denoting polarity.
• Variable Capacitor:
  • Symbol in (c).
  • Arrow diagonally crossing indicates tunability.

Figure 8.9: Circuit symbols for capacitors.

This slide is about circuit diagrams. It shows the standard symbols used to represent different types of capacitors in schematic diagrams. Understanding these symbols is crucial for reading and designing electronic circuits.

Application: Cell Membranes as Capacitors

• Biological Relevance: Cell membranes separate cells from surroundings, allowing selective ion passage.
• Potential Difference: Approximately 70 mV across a 7-10 nm thick membrane.
• Electric Field Strength: \[E = \frac{V}{d} = \frac{70 \times 10^{-3} \text{ V}}{10 \times 10^{-9} \text{ m}} = 7 \times 10^6 \text{ V/m} > 3 \text{ MV/m}\]
• Significance: This electric field strength is substantial (greater than air’s dielectric strength), highlighting the critical role of cell membranes in biological processes.
• “Fast Block”: Ernest Everett Just discovered that egg membranes undergo a depolarizing “wave of negativity” upon fertilization, preventing multiple sperm fusion.

Figure 8.10: Ion distribution across a cell membrane.

This section introduces a fascinating biological application of capacitors: the cell membrane. By treating the membrane as a tiny capacitor, we can calculate the enormous electric fields present across it, which are critical for cellular function and processes like fertilization. This connection demonstrates the broad applicability of physics concepts.

8.2 Capacitors in Series and in Parallel

Series Combination of Capacitors

• Arrangement: Capacitors connected end-to-end in a row.
• Charge: All capacitors acquire the same charge \(Q\).
  • Due to charge conservation (charge removed from one plate appears on the next).
• Voltage: Total voltage \(V\) is the sum of individual voltages (\(V = V_1 + V_2 + V_3\)). \(V_1 = Q/C_1\), \(V_2 = Q/C_2\), etc.
  
• Equivalent Capacitance (\(C_S\)): Defined by \(V = Q/C_S\). Substituting and canceling \(Q\): \[\frac{1}{C_S} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots\] (Equation 8.7)
• Characteristic: \(C_S\) is smaller than the smallest individual capacitance.

Figure 8.11: (a) Capacitors in series. (b) Equivalent capacitance.

When capacitors are connected in series, a few key rules apply. Most importantly, they all share the same charge, and the total voltage across the combination is the sum of individual voltages. The equivalent capacitance for a series combination is found by summing the reciprocals of individual capacitances. This leads to an equivalent capacitance that is always smaller than any of the individual capacitors, which is counterintuitive for resistors.

Parallel Combination of Capacitors

• Arrangement: One plate of each capacitor connected to one side of the circuit, other plates to the other side.
• Voltage: All capacitors have the same voltage \(V\) across their plates.
• Charge: Total charge \(Q\) stored is the sum of individual charges (\(Q = Q_1 + Q_2 + Q_3\)). \(Q_1 = C_1 V\), \(Q_2 = C_2 V\), etc.
  
• Equivalent Capacitance (\(C_P\)): Defined by \(Q = C_P V\). Substituting and canceling \(V\) gives
  \[ C_P = C_1 + C_2 + C_3 + \dots \tag{8.8} \]
  
• Characteristic: \(C_P\) is larger than any individual capacitance.

Figure 8.12: (a) Capacitors in parallel. (b) Equivalent capacitance.

In a parallel combination, all capacitors experience the same voltage, but the total charge is distributed among them. The equivalent capacitance is simply the sum of the individual capacitances. This means the parallel combination stores more charge for a given voltage, resulting in a larger equivalent capacitance than any single capacitor. This behavior is opposite to that of series capacitors and also opposite to how resistors combine in parallel.

Complex Capacitor Networks

• Networks often combine series and parallel connections.
• Strategy:
  1. Identify parts that are purely series or purely parallel.
  2. Calculate their equivalent capacitances.
  3. Replace the combination with its equivalent.
  4. Repeat until the equivalent capacitance of the entire network is found.

Figure 8.13: (a) A complex capacitor network. (b) Simplification to series. (c) Further simplification to parallel.

Real-world circuits rarely have just simple series or parallel combinations. This slide introduces the systematic approach to analyzing complex networks. By breaking down the network into simpler series and parallel parts, we can progressively reduce it to a single equivalent capacitance, making the analysis manageable.

Example 8.7: Network of Capacitors

Problem:

Determine the net capacitance \(C\) and the charge/voltage across each capacitor for the combination shown, with \(C_1 = 12.0\ \mu\text{F}\), \(C_2 = 2.0\ \mu\text{F}\), \(C_3 = 4.0\ \mu\text{F}\), and a \(12.0\ \text{V}\) potential difference across the network.

Solution:

1. C₂ and C₃ are in parallel: \(C_{23}=C_2+C_3=2.0\ \mu\text{F}+4.0\ \mu\text{F}=6.0\ \mu\text{F}\)

2. C₁ and C₂₃ are in series: \(\displaystyle \frac{1}{C}= \frac{1}{C_1}+ \frac{1}{C_{23}} = \frac{1}{12.0\ \mu\text{F}}+ \frac{1}{6.0\ \mu\text{F}} = \frac{1}{4.0\ \mu\text{F}}\), so \(C=4.0\ \mu\text{F}\)

Figure 8.14: (a) Original network. (b) Simplified network.

This example walks through a common scenario where both series and parallel combinations are present. We first simplify the parallel branch, then combine that equivalent capacitance with the series capacitor. The second part of the solution (on the next slide) delves into finding individual charges and voltages, which is often a crucial step in circuit analysis.

Example 8.7: Network of Capacitors (Cont.)

Solution (Cont.):

3. Charges and Voltages:

• \(C_1\) and \(C_{23}\) are in series, so \(Q_1 = Q_{23}\).
• Total voltage: \(12.0\ \text{V} = V_1 + V_{23} = \dfrac{Q_1}{C_1} + \dfrac{Q_1}{C_{23}}\).
• \(12.0\ \text{V} = \dfrac{Q_1}{12.0\ \mu\text{F}} + \dfrac{Q_1}{6.0\ \mu\text{F}} \;\Rightarrow\; Q_1 = 48.0\ \mu\text{C}\).
• \(V_1 = \dfrac{Q_1}{C_1} = \dfrac{48.0\ \mu\text{C}}{12.0\ \mu\text{F}} = 4.0\ \text{V}\).
• \(V_{23} = 12.0\ \text{V} - V_1 = 12.0\ \text{V} - 4.0\ \text{V} = 8.0\ \text{V}\).
• Since \(C_2\) and \(C_3\) are in parallel, \(V_2 = V_3 = V_{23} = 8.0\ \text{V}\).
• \(Q_2 = C_2 V_2 = (2.0\ \mu\text{F})(8.0\ \text{V}) = 16.0\ \mu\text{C}\).
• \(Q_3 = C_3 V_3 = (4.0\ \mu\text{F})(8.0\ \text{V}) = 32.0\ \mu\text{C}\).

Note

Notice that \(Q_2 + Q_3 = 16.0 \,\mu\text{C} + 32.0 \,\mu\text{C} = 48.0 \,\mu\text{C}\), which matches \(Q_1\) as expected for the series connection.

Here, we complete the example by calculating the individual charges and voltages across each capacitor. This requires careful application of the rules for series (same charge, voltages add) and parallel (same voltage, charges add) combinations. It’s a comprehensive problem that tests understanding of both equivalent capacitance and charge/voltage distribution.

8.3 Energy Stored in a Capacitor

Energy Storage in Capacitors

• Capacitors store electrostatic potential energy (\(U_C\)).
  
• Energy is stored in the electric field between its plates.
  
• When disconnected from a battery, energy remains in the field.
  
• Energy density (\(u_E\)): Energy per unit volume in free space with electric field \(E\).
  \[u_E = \frac{1}{2}\varepsilon_0 E^2\] (Equation 8.9)
  
• Total energy stored (\(U_C\)): For a parallel‑plate capacitor: \(U_C = u_E (A d) = \tfrac{1}{2}\varepsilon_0 E^2 A d\). Since \(E = V/d\) and \(C = \varepsilon_0 A/d\):
  \[U_C = \frac{1}{2} C V^2\]

Figure 8.15: Capacitors on a circuit board.

Capacitors are not just charge storage devices; they store energy in the electric field generated between their plates. This energy storage capability is what makes them useful in applications like camera flashes and defibrillators. The energy density formula is a general principle, and we can derive the total energy stored in a capacitor from it, or from considering the work done to charge the capacitor.

Forms of Energy Stored in a Capacitor

• The total work \(W\) required to charge a capacitor to a charge \(Q\) is the electrical potential energy \(U_C\) stored in it.
  
• \(dW = V\,dq = \frac{q}{C}\,dq\)
• \[W = \int_0^Q \frac{q}{C} dq = \frac{1}{2}\frac{Q^2}{C}\]
• Equivalent Forms (Equation 8.10): \[U_C = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV\]

Tip

These expressions are generally valid for all types of capacitors, not just parallel-plate.

This slide presents the different, but equivalent, ways to express the energy stored in a capacitor. The derivation from integrating the work done during charging emphasizes the physical process. These three forms are extremely useful, depending on which quantities (Q, V, C) are known in a given problem.

Application: Heart Defibrillator

• Device: Automated External Defibrillator (AED).
• Function: Delivers a large, short burst of electrical energy (a “shock”) to correct abnormal heart rhythms (arrhythmias).
• Mechanism: Energy is stored in a capacitor, then rapidly discharged.
• Clinical Relevance: Essential for treating cardiac arrest caused by ventricular fibrillation, allowing the heart’s natural pacemaker to resume normal rhythm.

Figure 8.16: Automated external defibrillator (AED).

The heart defibrillator is a powerful and life-saving application of capacitor energy storage. This slide connects the abstract concept of stored energy to a very tangible and high-impact medical device. It’s a great real-world example of how these physics principles are applied to save lives.

Example 8.9: Capacitance of a Heart Defibrillator

Problem:

A heart defibrillator delivers \(4.00 \times 10^{2}\ \text{J}\) of energy by discharging a capacitor initially at \(1.00 \times 10^{4}\ \text{V}\). What is its capacitance?

Strategy:

Given energy \(U_C\) and voltage \(V\), find \(C\) using \(U_C = \frac{1}{2} C V^{2}\).

Solution:

Rearranging the equation for \(C\):

\[C = \frac{2U_C}{V^2}\] \[C = \frac{2(4.00 \times 10^2 \text{ J})}{(1.00 \times 10^4 \text{ V})^2}\] \[C = \frac{800 \text{ J}}{1.00 \times 10^8 \text{ V}^2} = 8.00 \times 10^{-6} \text{ F} = 8.00 \text{ μF}\]

Note

This calculation demonstrates that even powerful medical devices utilize capacitance values in the microfarad range.

This example applies the energy formula to a practical scenario. It shows how to calculate the capacitance of a defibrillator given its energy output and operating voltage. This reinforces the relationship between energy, capacitance, and voltage and provides a concrete example of capacitor sizing in a critical application.

8.4 Capacitor with a Dielectric

Dielectrics and Capacitance

• Dielectric: An insulating material placed between capacitor plates.
• Effect on Capacitance: Increases capacitance.
• Experiment:
  1. Charge an empty capacitor (\(C_0\), \(V_0\), \(Q_0\)).
  2. Disconnect the battery (so \(Q_0\) remains constant).
  3. Insert a dielectric.
  4. Observe that the voltage drops to \(V < V_0\).
     • \(V = \frac{V_0}{\kappa}\)
     • \(\kappa\) is the dielectric constant (\(\kappa > 1\)).
• New Capacitance \(C\): Since \(Q\) is constant: \[C = \frac{Q₀}{V} = \frac{Q₀}{V₀/κ} = κ \frac{Q₀}{V₀} = κC₀\] (Equation 8.11)

Figure 8.17: Effect of inserting a dielectric into a charged capacitor.

This section introduces dielectrics – insulating materials placed between capacitor plates. Crucially, a dielectric increases a capacitor’s capacitance. We explain this with an experiment where a charged capacitor (disconnected from its battery) experiences a voltage drop upon dielectric insertion, while its charge remains constant. This voltage drop is quantified by the dielectric constant kappa, directly leading to the increase in capacitance.

Dielectrics and Stored Energy

• Energy \(U\) with a dielectric:
  • When a dielectric is inserted into an isolated, charged capacitor, the stored energy \(U\) decreases.
  • \[U = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}\frac{Q₀^2}{κC₀} = \frac{1}{κ} U₀\] (Equation 8.12)
• Why energy decreases:
  • The dielectric material is polarized by the capacitor’s electric field.
  • Induced charges on the dielectric surface are attracted to the free charges on the plates.
  • The dielectric is “pulled” into the gap, and work is done on the dielectric.
  • This work comes at the expense of the stored electrical energy.

Figure 8.18: Electronic stud finder using dielectric principles.

Not only does a dielectric change capacitance, but it also affects the stored energy. For an isolated (disconnected) charged capacitor, inserting a dielectric reduces the stored energy. This reduction is explained by the work done by the electric field on the polarizing dielectric material as it’s drawn into the capacitor, effectively “consuming” some of the stored energy.

Example 8.10: Inserting a Dielectric into an Isolated Capacitor

Problem:

An empty \(20.0\ \text{pF}\) capacitor is charged to \(40.0\ \text{V}\), and the battery is disconnected. A Teflon™ piece (\(\kappa = 2.1\)) is inserted. Find

• 1. \(C\)
     
• 2. \(Q\)
     
• 3. \(V\)
     
• 4. \(U\) with and without the dielectric.

Solution:

Given: \(C_0 = 20.0\ \text{pF}\), \(V_0 = 40.0\ \text{V}\), \(\kappa = 2.1\).

(a) Capacitance

\[ C = \kappa C_0 = 2.1 \times 20.0\ \text{pF} = 42.0\ \text{pF} \]

(b) Charge (battery disconnected, so \(Q\) is constant)

\[ Q_0 = C_0 V_0 = (20.0\ \text{pF})(40.0\ \text{V}) = 0.8\ \text{nC} \]

Thus \(Q = 0.8\ \text{nC}\) with the dielectric.

(c) Potential Difference

\[ V = \frac{V_0}{\kappa} = \frac{40.0\ \text{V}}{2.1} \approx 19.0\ \text{V} \]

(d) Energy Stored

Without dielectric:

\[ U_0 = \tfrac12 C_0 V_0^2 = \tfrac12 (20.0\ \text{pF})(40.0\ \text{V})^2 = 16.0\ \text{nJ} \]

With dielectric:

\[ U = \frac{U_0}{\kappa} = \frac{16.0\ \text{nJ}}{2.1} \approx 7.6\ \text{nJ} \]

Important

The dielectric significantly increases capacitance but decreases stored energy for an isolated capacitor.

This example quantitatively illustrates the effects of a dielectric on an isolated charged capacitor. It shows how capacitance increases, voltage decreases, and stored energy decreases, all by a factor of the dielectric constant kappa, while the charge remains constant. This is a critical scenario to understand.

8.5 Molecular Model of a Dielectric

Polarization at the Molecular Level

• Molecules: Can be polar (permanent dipole moment) or nonpolar (no permanent dipole).
  • Examples: Water is polar, oxygen is nonpolar.
• In an external electric field \(\vec{E}_0\):
  1. Polar molecules: Dipoles align with \(\vec{E}_0\) (randomly oriented without field).
  2. Nonpolar molecules: Undergo induced polarization; \(\vec{E}_0\) separates positive and negative charges, creating induced dipoles that align with \(\vec{E}_0\).
• Result: Net charge within the dielectric volume remains zero, but surface charges (\(+Q_i\), \(-Q_i\)) are induced on the dielectric faces adjacent to the capacitor plates.

Figure 8.19: Polarization of atoms/molecules.

To truly understand why dielectrics work, we need to look at the molecular level. This slide explains how both polar and nonpolar molecules respond to an external electric field by aligning or forming induced dipoles. This molecular alignment leads to the formation of surface charges on the dielectric, which is the key to its macroscopic effect on the electric field and capacitance.

Dielectric’s Effect on Electric Field

• Induced Electric Field (\(\vec{E}_i\)): The induced surface charges (\(+Q_i\), \(-Q_i\)) create their own electric field \(\vec{E}_i\) within the dielectric.
  
• \(\vec{E}_i\) opposes the external field \(\vec{E}_0\) (from free charges on capacitor plates).
  
• Net Electric Field (\(\vec{E}\)): \(\vec{E} = \vec{E}_0 + \vec{E}_i\) (Equation 8.13). Since \(\vec{E}_i\) opposes \(\vec{E}_0\), the magnitude \(E < E_0\).
  
• Dielectric Constant (\(\kappa\)): \(\kappa = \frac{E_0}{E}\) (Equation 8.14). \(\kappa > 1\) since \(E < E_0\).
  
• Induced Field in terms of \(\vec{E}_0\): \(\vec{E}_i = \left(\frac{1}{\kappa} - 1\right)\vec{E}_0\) (Equation 8.15).

Figure 8.20: Dielectric polarization and induced field.

Figure 8.21: Net electric field in a dielectric-filled capacitor.

The core consequence of dielectric polarization is the creation of an internal electric field E_i that opposes the external field E_0 from the capacitor plates. This reduces the net electric field within the dielectric, which in turn reduces the potential difference for a given charge, thereby increasing capacitance. The dielectric constant kappa is defined by this reduction in the electric field.

Dielectric Breakdown

• Definition: When the external electric field becomes too large, dielectric molecules ionize.
  • Electrons are removed, becoming free electrons.
  • The material becomes conductive, allowing charge to flow between plates.
• Dielectric Strength (\(E_c\)): The critical electric field value at which ionization (breakdown) occurs, imposing a limit on the voltage that can be applied across a capacitor.
  
• Example: Air’s dielectric strength is \(3.0\ \text{MV/m}\). For \(d = 1.0\ \text{mm}\),
  \[ V_{\text{max}} = E_c \, d = (3.0 \times 10^{6}\ \text{V/m})(1.0 \times 10^{-3}\ \text{m}) = 3.0\ \text{kV}. \]
  
• Teflon™: \(E_c \approx 60.0\ \text{MV/m}\). Allows for much higher voltages and stored charges.

Caution

Exceeding the dielectric strength can permanently damage the capacitor.

Dielectric breakdown is a critical limiting factor for capacitors. This slide defines what it is—the point where an insulator becomes a conductor due to a strong electric field—and introduces the concept of dielectric strength. Understanding this limit is vital for safe and effective capacitor design and use, as it dictates the maximum voltage a capacitor can withstand.

Table of Dielectric Constants and Strengths

Material	Dielectric constant κ	Dielectric strength E_c [×10⁶ V/m]
Vacuum	1	∞
Dry air (1 atm)	1.00059	3.0
Teflon™	2.1	60 to 173
Paraffin	2.3	11
Silicone oil	2.5	10 to 15
Polystyrene	2.56	19.7
Nylon	3.4	14
Paper	3.7	16
Fused quartz	3.78	8
Glass	4 to 6	9.8 to 13.8
Concrete	4.5	–
Bakelite	4.9	24
Diamond	5.5	2,000
Pyrex glass	5.6	14
Mica	6.0	118
Neoprene rubber	6.7	15.7 to 26.7
Water	80	––
Sulfuric acid	84 to 100	––
Titanium dioxide	86 to 173	–
Strontium titanate	310	8
Barium titanate	1,200 to 10,000	–
Calcium copper titanate	> 250,000	–

Table 8.1: Representative values for various materials.

This table provides a useful reference for dielectric constants and dielectric strengths of various common materials. It highlights the wide range of properties available and demonstrates that not all materials with high dielectric constants are good insulators, such as water, due to the presence of free ions.

Example 8.12: Dielectric in a Capacitor Connected to a Battery

Problem:

An empty capacitor (\(C_0\), \(V_0\), \(E_0\), \(Q_0\)) is connected to a battery (\(V_0\)). A dielectric (\(\kappa\)) is inserted while the battery remains connected. Find
(a) \(C\), \(V\), \(E\) and
(b) \(Q\) and \(Q_i\) (induced charge) in terms of \(Q_0\).

Solution:

1. Capacitance: \(C = \kappa C_{0}\). Voltage: Battery remains connected, so \(V\) is constant: \(V = V_{0}\). Electric Field: Since \(V = E d\) and \(d\) is constant, \(E = V/d = V_{0}/d = E_{0}\).

2. Free charge \(Q\): \(Q = C V = (\kappa C_{0}) V_{0} = \kappa (C_{0} V_{0}) = \kappa Q_{0}\). Induced charge \(Q_{i}\):

The net field \(E\) is due to the effective charge \((Q - Q_{i})\).

Since \(E = E_{0}\), then \(\frac{Q - Q_{i}}{\varepsilon_{0} A} = \frac{Q_{0}}{\varepsilon_{0} A}\), so \(Q - Q_{i} = Q_{0}\).

Hence \(Q_{i} = Q - Q_{0} = \kappa Q_{0} - Q_{0} = (\kappa - 1) Q_{0}\).

Note

When the battery remains connected, the voltage is constant. The charge on the plates increases by a factor of κ, and the induced charge Q_i accounts for this increase in storage capacity. The stored energy increases in this case as U = κU₀.

This example contrasts with Example 8.10 by considering a capacitor that remains connected to a battery during dielectric insertion. In this case, the voltage is held constant, leading to an increase in charge on the plates. We calculate how capacitance increases, and how the free and induced charges are related. It’s important to distinguish between these two scenarios: isolated vs. connected to a battery.

Key Takeaways

• Capacitors: Store charge and energy, defined by \(C = Q/V\).
• Capacitance depends on geometry:
  • Parallel‑plate: \(C = \varepsilon_0 A/d\)
  • Spherical: \(C = 4\pi\varepsilon_0 \frac{R_1 R_2}{R_2 - R_1}\)
  • Cylindrical: \(C = 2\pi\varepsilon_0 \, l / \ln(R_2/R_1)\)
• Combinations:
  • Series: \(1/C_S = \sum_i (1/C_i)\) (same \(Q\), \(V\) adds)
  • Parallel: \(C_P = \sum_i C_i\) (same \(V\), \(Q\) adds)
• Energy storage: \(U_C = \tfrac12 C V^2 = \tfrac12 Q^2 / C = \tfrac12 Q V\). Energy density \(u_E = \tfrac12 \varepsilon_0 E^2\).
• Dielectrics: Insulating materials that increase capacitance \(C = \kappa C_0\).
  • Reduce electric field \(E = E_0/\kappa\) and voltage \(V = V_0/\kappa\) (for isolated capacitor).
  • Reduce stored energy \(U = U_0/\kappa\) (for isolated capacitor).
  • Increase stored energy \(U = \kappa U_0\) (for capacitor connected to battery).
• Molecular Model: Dielectrics polarize, creating an opposing \(E_i\) field, reducing net \(E\).
• Dielectric Breakdown: Occurs when electric field exceeds dielectric strength \(E_c\), making material conductive.

This slide summarizes the most important concepts and formulas covered in this chapter. It’s designed for a quick review and to highlight the essential information students should retain.

Key Equations

Equation	Description
\(C = Q/V\)	Definition of Capacitance
\(C = \varepsilon_0 A/d\)	Capacitance of a Parallel‑Plate Capacitor
\(C = \dfrac{4\pi\varepsilon_0 R_1 R_2}{R_2 - R_1}\)	Capacitance of a Spherical Capacitor
\(C = \dfrac{2\pi\varepsilon_0 \, l}{\ln(R_2/R_1)}\)	Capacitance of a Cylindrical Capacitor
\(\dfrac{1}{C_S} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dots\)	Capacitors in Series
\(C_P = C_1 + C_2 + \dots\)	Capacitors in Parallel
\(U_C = \tfrac{1}{2} C V^2 = \tfrac{1}{2} \dfrac{Q^2}{C} = \tfrac{1}{2} Q V\)	Energy Stored in a Capacitor
\(u_E = \tfrac{1}{2} \varepsilon_0 E^2\)	Electric Field Energy Density
\(C = \kappa C_0\)	Capacitance with a Dielectric
\(V = V_0/\kappa\) (isolated) or \(V = V_0\) (connected)	Voltage with a Dielectric
\(E = E_0/\kappa\) (isolated) or \(E = E_0\) (connected)	Electric Field with a Dielectric
\(U = U_0/\kappa\) (isolated) or \(U = \kappa U_0\) (connected)	Energy with a Dielectric
\(E_c\)	Dielectric Strength

This table provides a concise list of all the key equations discussed in the chapter, along with their descriptions. It serves as a handy reference for students when solving problems or reviewing the material.

Key Terms

Term	Definition
Capacitor	A device used to store electrical charge and electrical energy.
Capacitance (C)	The ratio of the maximum charge \(Q\) that can be stored in a capacitor to the applied voltage \(V\) across its plates (\(C = Q/V\)).
Farad (F)	The SI unit of capacitance, equal to one coulomb per volt (\(1\ \text{C/V}\)).
Dielectric	An insulating material placed between the plates of a capacitor to increase its capacitance.
Dielectric Constant (κ)	A dimensionless factor by which capacitance increases when a dielectric fills the space between plates (\(C = κC_0\)). Also \(κ = E_0/E\).
Dielectric Breakdown	The phenomenon where an insulating material becomes conductive when the applied electric field exceeds its dielectric strength.
Dielectric Strength (E_c)	The critical value of the electric field at which dielectric breakdown occurs in an insulating material.
Series Combination	An arrangement of capacitors where they are connected end‑to‑end; all have the same charge.
Parallel Combination	An arrangement of capacitors where corresponding plates are connected together; all have the same voltage.
Energy Density (u_E)	The amount of energy stored per unit volume in an electric field: \[u_E = \frac{1}{2}\varepsilon_0 E^2\].
Polarization	The process by which an external electric field induces or aligns electric dipole moments in a dielectric material.
Permittivity of Free Space (ε₀)	A fundamental physical constant representing the absolute dielectric permittivity of a vacuum (\(8.85 \times 10^{-12}\ \text{F/m}\)).

This glossary provides definitions for all the important terms introduced in Chapter 8. It’s a useful tool for students to quickly grasp the vocabulary and concepts.