Detailed Notes from Toppers on Cylindrical and Spherical Capacitors- Series and Parallel Combinations

Capacitance of a Cylindrical Capacitor:

• Reference: NCERT Class 12, Chapter 4: Capacitance and Capacitors, Page 144

• The capacitance of a cylindrical capacitor is given by:

$$C = \frac{2\pi\varepsilon L}{\ln(b/a)}$$

• Where:

• C is the capacitance in farads (F)

• ε is the permittivity of the medium between the plates (F/m)

• L is the length of the capacitor in meters (m)

• a is the radius of the inner cylinder in meters (m)

• b is the radius of the outer cylinder in meters (m)

• The capacitance of a cylindrical capacitor is directly proportional to the length of the capacitor and inversely proportional to the natural logarithm of the ratio of the radii of the inner and outer cylinders.

Capacitance of a Spherical Capacitor:

• Reference: NCERT Class 12, Chapter 4: Capacitance and Capacitors, Page 146

• The capacitance of a spherical capacitor is given by:

$$C = 4\pi\varepsilon_{0} \frac{ab}{b-a}$$

• Where:

• C is the capacitance in farads (F)

• ε₀ is the vacuum permittivity (8.85 × 10^-12 F/m)

• a is the radius of the inner sphere in meters (m)

• b is the radius of the outer sphere in meters (m)

• The capacitance of a spherical capacitor is directly proportional to the radii of the inner and outer spheres and inversely proportional to the distance between the spheres.

Capacitors in Series:

• Reference: NCERT Class 12, Chapter 4: Capacitance and Capacitors, Page 152

• When capacitors are connected in series, the equivalent capacitance is given by:

$$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + …$$

• Where:

• C_eq is the equivalent capacitance in farads (F)

• C₁, C₂, C₃,… are the capacitances of the individual capacitors in farads (F)

• The voltage across each capacitor in a series combination is inversely proportional to its capacitance.

Capacitors in Parallel:

• Reference: NCERT Class 12, Chapter 4: Capacitance and Capacitors, Page 153

• When capacitors are connected in parallel, the equivalent capacitance is given by:

$$C_{eq} = C_1 + C_2 + C_3 + …$$

• Where:

• C_eq is the equivalent capacitance in farads (F)

• C₁, C₂, C₃,… are the capacitances of the individual capacitors in farads (F)

• The charge on each capacitor in a parallel combination is directly proportional to its capacitance.

Series and Parallel Combinations of Cylindrical and Spherical Capacitors:

• By combining the concepts of series and parallel connections, it is possible to analyze combinations of cylindrical and spherical capacitors connected in various configurations.

• The equivalent capacitance and voltage/charge distribution in such combinations can be determined using the appropriate formulas.

• Numerical problems involving cylindrical and spherical capacitors can be solved by applying these concepts.

Energy Stored in Cylindrical and Spherical Capacitors:

• Reference: NCERT Class 12, Chapter 4: Capacitance and Capacitors, Page 155

• The energy stored in a cylindrical capacitor is given by:

$$U = \frac{1}{2}CV^2$$

• Where:

• U is the energy stored in joules (J)

• C is the capacitance in farads (F)

• V is the voltage across the capacitor in volts (V)

• The energy stored in a spherical capacitor is given by:

$$U = \frac{1}{2}\frac{QV^2}{C}$$

• Where:

• U is the energy stored in joules (J)

• Q is the charge stored on the capacitor in coulombs (C)

• C is the capacitance in farads (F)

• V is the voltage across the capacitor in volts (V)

• The energy stored in a capacitor is directly proportional to the capacitance and the square of the voltage or charge.

Practical Applications of Capacitors:

• Cylindrical and spherical capacitors find applications in various electronic devices, including:

• Energy storage devices

• Filters

• Timing circuits

• Tuning circuits

• Amplifiers

• Power supplies

• The choice of capacitor type (cylindrical or spherical) depends on factors such as capacitance value, voltage rating, size, and cost.

• Capacitors are essential components in electronic circuits and play a vital role in controlling and storing electrical energy.

By diligently studying these subtopics, referring to the NCERT textbooks, and consistently solving practice problems, aspirants can enhance their understanding of Cylindrical and Spherical Capacitors- Series and Parallel Combinations, thereby increasing their chances of excelling in the JEE exam.