Shortcut Methods and Tricks for LC Oscillations

JEE Mains

1. Resonance Formula

$$f_0=\frac{1}{2\pi\sqrt{LC}}$$

Where:

• $$f_0$$ is the resonant frequency in Hz
• $$L$$ is the inductance in Henry
• $$C$$ is the capacitance in Farads

2. Time Period Formula

$$T_0=\frac{2\pi\sqrt{LC}}{1}$$

Where:

• $$T_0$$ is the time period in seconds
• $$L$$ is the inductance in Henry
• $$C$$ is the capacitance in Farads

3. Energy Storage Formula

The total energy stored in an LC circuit at any instant is given by:

$$E=\frac{1}{2}LI^2+\frac{1}{2}CV^2$$

Where:

• $$E$$ is the total energy stored in Joules
• $$L$$ is the inductance in Henry
• $$I$$ is the current in Amperes
• $$C$$ is the capacitance in Farads
• $$V$$ is the voltage in Volts

4. Quality Factor (Q) Formula

The quality factor (Q) of an LC circuit is a measure of its efficiency. It is given by:

$$Q=\frac{\omega L}{R}$$

Where:

• $$Q$$ is the quality factor
• $$\omega$$ is the angular frequency
• $$L$$ is the inductance in Henries
• $$R$$ is the resistance in Ohms

CBSE Boards

1. Resonance Formula

$$f_0=\frac{1}{2\pi\sqrt{LC}}$$

Where:

• $$f_0$$ is the resonant frequency in Hertz
• $$L$$ is the inductance in Henry
• $$C$$ is the capacitance in Farads

2. Time Period Formula

$$T_0=\frac{1}{f_0}$$

Where:

• $$T_0$$ is the time period in seconds
• $$f_0$$ is the resonant frequency in Hertz

3. Energy Storage Formula

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

Where:

• $$E$$ is the energy stored in Joules
• $$C$$ is the capacitance in Farads
• $$V$$ is the voltage across the capacitor in Volts

4. Quality Factor (Q) Formula

$$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$$

Where:

• $$Q$$ is the quality factor
• $$R$$ is the resistance of the inductor in ohms
• $$L$$ is the inductance of the inductor in Henries
• $$C$$ is the capacitance of the capacitor in Farads