JEE ADVANCED Capacitive circuits and alternating currents

Shortcut methods and tricks

• Capacitor charging and discharging:

• Use the time constant (\tau = RC) to determine the rate of charging and discharging.

• The voltage across a capacitor during charging is given by $$V_c(t)=V_0(1-e^{-\frac{t}{RC}}).$$

• AC circuit analysis:

• Use the impedance formula (Z = \sqrt{R^2 + (X_L - X_C)^2}) to calculate the total impedance of a circuit.

• The phase angle (\phi ) is given by (\phi = \tan^{-1} \frac{X_L - X_C}{R}).

• The power factor is given by (pf = \cos(\phi)).

• Resonance in AC circuits:

• The resonant frequency of a series RLC circuit is given by $$f_0=\frac{1}{2\pi \sqrt{LC}}$$.

• At resonance, the impedance of a series RLC circuit is minimum and the current is maximum.

• Power in AC circuits:

• The average power dissipated in a resistor is given by $$P_{av}=\frac{1}{2}VI\cos \varphi$$.

• The power factor is a measure of how efficiently a circuit converts electrical energy into useful work. A power factor close to 1 indicates efficient conversion.

• RMS and average values:

• The RMS (root-mean-square) value, voltage, or current is given by $$V_{RMS}=\sqrt{\frac{1}{T}{\int }_0^T{v}^2(t)dt}$$ and $$I_{RMS}=\sqrt{\frac{1}{T}{\int }_0^T{i}^2(t)dt}$$ where $$T$$ is the time period of the waveform.

• The average value of a voltage or current is given by $$V_{av}=\frac{1}{T}{\int }_0^{T}v(t)dt$$ and $$I_{av}=\frac{1}{T}{\int }_0^{T}i(t)dt$$

CBSE BOARD EXAMS Capacitive circuits and alternating currents

Shortcut methods and tricks

• Capacitor charging and discharging:

• Use the time constant (\tau = RC) to determine the rate of charging and discharging.

• The voltage across a capacitor during charging is given by (V_c(t) = V_0(1 - e^{-\frac{t}{RC}})).

• Capacitance and energy storage:

• The capacitance of a capacitor is given by $$C=\frac{Q}{V}$$, where Q is the charge stored on the capacitor and V is the voltage across it.

• The energy stored in a capacitor is given by $$E=\frac{1}{2}C{V}^2$$.

• AC circuit components:

• Resistors: Oppose the flow of current, causing the voltage and current to be in phase.

• Capacitors: Store electrical energy and cause the voltage and current to be out of phase.

• Inductors: Oppose changes in current flow and cause the voltage and current to be out of phase.

• Alternating current characteristics:

• Frequency: The number of cycles per second of an alternating current.

• Period: The time taken for one complete cycle of an alternating current.

• Amplitude: The maximum value of an alternating current.

• Phase difference:

• The voltage and current in a capacitive circuit are out of phase by 90 degrees.

• The voltage and current in an inductive circuit are out of phase by 180 degrees.

• Simple circuit calculations:

• Use Ohm’s law, (V = IR ), to calculate the voltage, current, and resistance in an AC circuit.

• Use the Pythagorean theorem to calculate the impedance of a circuit with both resistance and reactance.

• Use the power factor formula, $$pf=\frac{R}{Z}$$, to calculate the power factor of a circuit.