Notes from Toppers
Cylindrical and Spherical Capacitors
1. Cylindrical Capacitors:
-
NCERT Reference: Chapter 2: Electrostatics, Class 12
-
Keynotes:
- Capacitance of a cylindrical capacitor: $$C=\frac{2\pi\varepsilon_0 L}{\ln(b/a)}$$ where $$a, b$$ are the radii of the inner and outer cylinders, respectively, $$L$$ is the length of the cylinder, and $$\varepsilon_0$$ is the vacuum permittivity.
-
Effect of radii: Capacitance increases with increasing radii of both the inner and outer cylinders.
-
Potential difference: $$V_{ab}=\frac{Q}{4\pi\varepsilon_0 L}\ln\left(\frac{b}{a}\right)$$, where $$Q$$ is the charge on the capacitor and $$V_{ab}$$ is the potential difference between its terminals.
2. Spherical Capacitors:
-
NCERT Reference: Chapter 2: Electrostatics, Class 12
-
Keynotes:
- Capacitance of a spherical capacitor: $$C=4\pi\varepsilon_0\frac{b a}{b-a}$$ where $$a, b$$ are the radii of the inner and outer spheres, respectively, and $$\varepsilon_0$$ is the vacuum permittivity.
-
Effect of radii: Capacitance increases as the radius of the outer sphere increases and decreases as the radius of the inner sphere increases.
-
Potential difference: $$V_{ab}=\frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{a}-\frac{1}{b}\right)$$ where $$Q$$ is the charge on the capacitor and $$V_{ab}$$ is the potential difference between its terminals.
3. Series and Parallel Combinations of Capacitors:
-
NCERT Reference: Chapter 2: Electrostatics, Class 12
-
Series Combination:
- Equivalent capacitance: $$C_{eq}=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}+\cdots+\frac{1}{C_n}}$$
- Total charge: $$Q=C_{eq}V$$
- Voltage distribution: $$V_1=\frac{Q}{C_1}, V_2=\frac{Q}{C_2}, \cdots, V_n=\frac{Q}{C_n}$$ $$Total\space voltage: \sum V_i$$
-
Parallel Combination:
- Equivalent capacitance: $$C_{eq}=C_1+C_2+\cdots+C_n$$
- Total charge: $$Q=C_{eq}V$$
- Voltage distribution: $$V=V_1=V_2=\cdots=V_n$$
4. Energy Stored in Capacitors:
-
NCERT Reference: Chapter 2: Electrostatics, Class 12
-
Keynotes:
- Energy stored in a capacitor: $$U=\frac{1}{2}CV^2$$ where $$C$$ is the capacitance and $$V$$ is the voltage across the capacitor.
- Energy density: $$u=\frac{1}{2}\varepsilon E^2$$
- Factors affecting stored energy: capacitance, voltage, and electric field strength.
5. Applications of Cylindrical and Spherical Capacitors:
- Practical applications in electronic devices and circuits, such as:
- Energy storage in electronic flash units, defibrillators, and power systems
- Filtering in audio and radio circuits
- Tuning circuits in radios and televisions
6. Solved Examples and Numerical Problems:
- Numerical problems and theoretical questions involving cylindrical and spherical capacitors, series and parallel combinations, with step-by-step solutions.
Remember: Practicing with a variety of problems and consistently reviewing the concepts will strengthen understanding and enhance problem-solving skills in preparation for the JEE exam.