Cylindrical And Spherical Capacitors, Series And Parallel Combinations

Slide 1 - Cylindrical and Spherical Capacitors

Capacitors are devices used to store electric charge.
Cylindrical capacitors have a cylindrical geometry with inner and outer conductive cylinders separated by a dielectric medium.
Spherical capacitors have a spherical geometry with a central conducting sphere surrounded by an outer conducting shell.
The capacitance of a cylindrical capacitor is given by the formula: $ C = \frac{2\pi \varepsilon_0 L}{\ln\left(\frac{b}{a}\right)} $
The capacitance of a spherical capacitor is given by the formula: $ C = 4\pi\varepsilon_0 \left(\frac{1}{a} - \frac{1}{b}\right) $

Slide 2 - Series and Parallel Capacitor Combinations

Capacitors can be connected in series or parallel to effectively increase or decrease the overall capacitance.
In a series combination of capacitors, the total capacitance is given by: $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots + \frac{1}{C_n} $
In a parallel combination of capacitors, the total capacitance is the sum of individual capacitances: $ C_{\text{total}} = C_1 + C_2 + \ldots + C_n $
Series combinations reduce the total capacitance, while parallel combinations increase it.
Different combinations of capacitors can be used in electronic circuits to achieve specific functions.

Slide 3 - Example: Series Capacitor Combination

Suppose we have two capacitors connected in series, with capacitances $ C_1 = 5 \mu F $ and $ C_2 = 10 \mu F $ .
The total capacitance in this series combination can be calculated as: $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} $ $ \frac{1}{C_{\text{total}}} = \frac{1}{5 \mu F} + \frac{1}{10 \mu F} $
Simplifying the expression, we find: $ \frac{1}{C_{\text{total}}} = \frac{1}{10 \mu F} $
Therefore, the total capacitance in this series combination is $ C_{\text{total}} = 10 \mu F $ .

Slide 4 - Example: Parallel Capacitor Combination

Suppose we have two capacitors connected in parallel, with capacitances $ C_1 = 5 \mu F $ and $ C_2 = 10 \mu F $ .
The total capacitance in this parallel combination can be calculated as: $ C_{\text{total}} = C_1 + C_2 $ $ C_{\text{total}} = 5 \mu F + 10 \mu F $
Simplifying the expression, we find: $ C_{\text{total}} = 15 \mu F $
Therefore, the total capacitance in this parallel combination is $ C_{\text{total}} = 15 \mu F $ .

Slide 5 - Capacitors in Real-Life Applications

Capacitors find applications in various electronic devices and circuits.
They are used for energy storage, smoothing signals, and filtering in power supply circuits.
Capacitors are also used in timing circuits, oscillators, and filters for audio and radio frequency signals.
They play a crucial role in electronic equipment like computers, televisions, and mobile phones.
Capacitors enable the functioning of many electrical systems we use in our daily lives.

Slide 6 - Homework Problem

Consider a series combination of three capacitors with capacitances $ C_1 = 3 \mu F $ , $ C_2 = 6 \mu F $ , and $ C_3 = 9 \mu F $ .
Calculate the total capacitance in this series combination using the formula: $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} $
Solve the problem and submit your answer by the next class for evaluation.

Slide 7 - Recap

Capacitors are devices used to store electric charge.
Cylindrical capacitors have a cylindrical geometry, while spherical capacitors have a spherical geometry.
The capacitance of a cylindrical capacitor is given by \ $ C = \frac{2\pi \varepsilon_0 L}{\ln\left(\frac{b}{a}\right)}\ $ .
The capacitance of a spherical capacitor is given by \ $ C = 4\pi\varepsilon_0 \left(\frac{1}{a} - \frac{1}{b}\right)\ $ .
Capacitors can be connected in series or parallel to alter the overall capacitance.

Slide 8 - Recap (contd.)

In a series combination of capacitors, the inverse of the total capacitance is the sum of inverses of individual capacitances: \ $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots + \frac{1}{C_n}\ $ .
In a parallel combination of capacitors, the total capacitance is the sum of individual capacitances: \ $ C_{\text{total}} = C_1 + C_2 + \ldots + C_n\ $ .
Capacitors have various applications in electronic circuits and devices.
Solving problems involving capacitor combinations helps develop conceptual understanding.

Slide 9 - Questions?

Slide 10 - Thank You!

Stay curious, keep learning, and best wishes for your studies.
See you in the next class!

Slide 11 - Electric Field inside a Capacitor

The electric field inside a capacitor is constant and parallel to the plates.
It is given by the formula: $ E = \frac{V}{d} $ where $ V $ is the voltage across the plates and $ d $ is the separation between the plates.
The electric field lines are directed from the positive plate to the negative plate.
The electric field exerts a force on charged particles, causing them to move.

Slide 12 - Energy Stored in a Capacitor

A capacitor stores energy in the form of electric field.
The energy stored in a capacitor is given by the formula: $ U = \frac{1}{2}CV^2 $ where $ C $ is the capacitance of the capacitor and $ V $ is the voltage across it.
The energy stored is directly proportional to the square of the voltage and capacitance.

Slide 13 - Dielectric Material in a Capacitor

Dielectric materials are insulators used to separate the plates of a capacitor.
They increase the capacitance of a capacitor by reducing the electric field between the plates.
The presence of a dielectric material enhances the ability of the capacitor to store charge.
The effect of the dielectric material is characterized by its relative permittivity ( $ \varepsilon_r $ ).
The capacitance of a capacitor with a dielectric material is given by: $ C = \varepsilon_r C_0 $ where $ C_0 $ is the capacitance without dielectric.

Slide 14 - Dielectric Strength and Breakdown

Dielectric strength is the maximum electric field that a dielectric material can withstand without breaking down.
If the electric field exceeds the dielectric strength, the dielectric may experience a breakdown, leading to current flow.
The breakdown of a dielectric material can result in permanent damage to the capacitor or the surrounding circuit.
It is important to select a dielectric material with a high dielectric strength for safe and reliable operation.

Slide 15 - Capacitor Charging and Discharging

Capacitors can be charged by connecting them to a voltage source.
During charging, the capacitor accumulates electric charge and the voltage across it increases.
The time taken for a capacitor to charge or discharge to a certain percentage of its final voltage is given by the time constant ( $ \tau $ ).
The time constant is the product of the resistance ( $ R $ ) in the circuit and the capacitance ( $ C $ ) of the capacitor: $ \tau = RC $

Slide 16 - RC Circuits

RC circuits are circuits that contain resistors and capacitors.
These circuits are used in various applications such as timing circuits, filters, and oscillators.
The time constant ( $ \tau $ ) determines the behavior of the circuit. It represents the time it takes for the capacitor to charge or discharge.
The voltage across the capacitor in an RC circuit can be calculated using the formula: $ V(t) = V_0(1 - e^{-\frac{t}{\tau}}) $ where $ V(t) $ is the voltage at time $ t $ , $ V_0 $ is the initial voltage across the capacitor, and $ e $ is the logarithmic base.

Slide 17 - Example: RC Circuit

Consider an RC circuit with a resistor ( $ R $ ) of 1k $ \Omega $ and a capacitor ( $ C $ ) of 10 $ \mu F $ .
If the circuit is connected to a 10V power supply, calculate the voltage across the capacitor after 2 seconds.
Using the formula $ V(t) = V_0(1 - e^{-\frac{t}{\tau}}) $ , we can calculate the voltage as: $ V(2s) = 10V(1 - e^{-\frac{2s}{RC}}) $

Slide 18 - Applications of Capacitors

Capacitors have a wide range of applications in electronic circuits and devices.
They are used in power supply circuits to filter out unwanted noise and stabilize voltage.
Capacitors are used in timing circuits to control the frequency and duration of signals.
In audio systems, capacitors are used in filters to block or pass certain frequency components.
Many electronic components, such as transistors and integrated circuits, require capacitors for proper functioning.

Slide 19 - Capacitor Failure

Capacitors can fail due to various reasons, leading to a loss of functionality in the circuit.
Common reasons for capacitor failure include overheating, voltage spikes, and manufacturing defects.
Capacitors can fail in different ways, such as short-circuiting, open-circuiting, or losing capacitance.
Regular testing and inspection of capacitors can help identify and prevent failure in electronic systems.

Slide 20 - Summary

Capacitors are devices used to store electric charge.
The electric field inside a capacitor is constant and parallel to the plates.
Energy is stored in a capacitor, and the amount is given by $ U = \frac{1}{2}CV^2 $ .
Dielectric materials increase the capacitance of a capacitor.
Capacitors can be charged and discharged in RC circuits.
Capacitors have numerous applications but can also fail if not properly maintained.

Slide 21 - Electric Potential Energy of a Capacitor

The electric potential energy ( $ PE $ ) stored in a capacitor is given by the formula: $ PE = \frac{1}{2}CV^2 $
$ C $ is the capacitance of the capacitor and $ V $ is the voltage across it.
The potential energy is directly proportional to the square of the voltage and capacitance.
When a capacitor is fully charged, its potential energy is maximum.
This energy can be released to perform useful work in electronic devices.

Slide 22 - Capacitors in Series Combination

In a series combination of capacitors, the total capacitance is less than the individual capacitances.
The inverse of the total capacitance ( $ C_{\text{total}} $ ) is given by the sum of inverses of individual capacitances ( $ C_1 $ , $ C_2 $ , etc.): $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots + \frac{1}{C_n} $
The charges on capacitors in series combination are the same.
The voltages across individual capacitors are inversely proportional to their capacitances.

Slide 23 - Capacitors in Parallel Combination

In a parallel combination of capacitors, the total capacitance is the sum of individual capacitances.
The total capacitance ( $ C_{\text{total}} $ ) is given by the sum of individual capacitances ( $ C_1 $ , $ C_2 $ , etc.): $ C_{\text{total}} = C_1 + C_2 + \ldots + C_n $
The voltage across capacitors in parallel combination is the same.
The charges on individual capacitors are proportional to their capacitances.

Slide 24 - Example: Capacitors in Series Combination

Suppose we have three capacitors in series with capacitances $ C_1 = 2 \mu F $ , $ C_2 = 4 \mu F $ , and $ C_3 = 6 \mu F $ .
The inverse of the total capacitance in this series combination can be calculated as: $ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} $
Simplifying the expression, we find: $ \frac{1}{C_{\text{total}}} = \frac{1}{2 \mu F} + \frac{1}{4 \mu F} + \frac{1}{6 \mu F} $
Therefore, the total capacitance in this series combination is $ C_{\text{total}} = \frac{3}{7} \mu F $ .

Slide 25 - Example: Capacitors in Parallel Combination

Suppose we have three capacitors in parallel with capacitances $ C_1 = 2 \mu F $ , $ C_2 = 4 \mu F $ , and $ C_3 = 6 \mu F $ .
The total capacitance in this parallel combination can be calculated as: $ C_{\text{total}} = C_1 + C_2 + C_3 $
Simplifying the expression, we find: $ C_{\text{total}} = 2 \mu F + 4 \mu F + 6 \mu F $
Therefore, the total capacitance in this parallel combination is $ C_{\text{total}} = 12 \mu F $ .

Slide 26 - Energy Dissipated in a Resistor

When a capacitor discharges through a resistor, energy is dissipated in the form of heat.
The energy dissipated ( $ E_{\text{dissipated}} $ ) can be calculated using the formula: $ E_{\text{dissipated}} = \frac{1}{2}CV^2\left(1 - e^{-\frac{t}{RC}}\right) $
$ C $ is the capacitance, $ V $ is the initial voltage, $ R $ is the resistance, and $ t $ is the time.
The energy dissipated depends on the time constant ( $ \tau $ ), which is the product of $ R $ and $ C $ .

Slide 27 - Charging and Discharging of a Capacitor

When a capacitor is connected to a voltage source, it charges until the voltage across it reaches the source voltage.
The time taken for a capacitor to charge to approximately 63.2% of the source voltage is given by the time constant ( $ \tau $ ).
Similarly, when a charged capacitor is disconnected from the source, it discharges over time with a similar time constant.
The time constant represents the time it takes for the voltage across the capacitor to increase or decrease by approximately 63.2% of the difference between the initial and final voltages.

Slide 28 - Example: Charging and Discharging of a Capacitor

Suppose we have a capacitor with a capacitance of $ C = 10 \mu F $ and a resistor of $ R = 100 \Omega $ .
If the capacitor is charged with a voltage of $ V_0 = 10V $ , the time constant can be calculated as: $ \tau = RC = (100 \Omega)(10 \mu F) $
The time constant is $ \tau = 1ms $ .
After one time constant ( $ \tau $ ) of charging, the voltage across the capacitor reaches approximately 63.2% of the source voltage.
Similarly, after one time constant ( $ \tau $ ) of discharging, the voltage across the capacitor decreases to approximately 36.8% of the initial voltage.

Slide 29 - Applications of Capacitors

Capacitors have various applications in electronic circuits and devices.
They can be used in power supply circuits to smooth out voltage fluctuations.
Capacitors are also used in tuning circuits to select specific frequencies in radio receivers.
They are essential components in data storage devices, such as hard drives and flash memory.
Capacitors are used in timing circuits for precise control of electronic signals.
They are also used in filtering circuits to block unwanted frequencies.

Slide 30 - Summary

Capacitors store electric charge and have various applications in electronics.
Series combinations of capacitors have reduced total capacitance and proportional charge distribution.
Parallel combinations of capacitors have increased total capacitance and equal voltage distribution.
The energy stored in a capacitor is given by $ PE = \frac{1}{2}CV^2 $ .
Capacitors can charge and discharge in RC circuits with a time constant of $ \tau = RC $ .
Capacitors find use in power supply circuits, tuning circuits, timing circuits, data storage devices, and more.