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Chapter 7: Alternating Current - Medium / Standard (Level 2)
Topic 7.1: Average and RMS Values
1.
Derive the mathematical expression for the mean (average) value of a sinusoidal alternating current ($I = I_0 \sin\omega t$) over the positive half cycle.
2.
An alternating current is given by the equation $I = 3 + 4\sin(\omega t)$. Find the RMS value of this current.
3.
Why is the scale of a standard hot-wire AC ammeter non-uniform (crowded at the beginning and spread out at the end)?
4.
Calculate the exact time taken by an alternating current of frequency $50\text{ Hz}$ to reach from zero to its RMS value for the first time.
5.
Find the RMS value of a symmetrical square wave current alternating periodically between a steady positive peak $+I_0$ and a steady negative peak $-I_0$.
6.
What is the mean value of a full-wave rectified AC voltage having a peak value $V_0$? Compare it with the half-wave rectified case.
7.
Can a standard DC moving-coil ammeter measure the RMS value of an AC current if we just apply a correction factor? Explain.
8.
An AC voltage is given by $V = 100\sqrt{2} \sin(100\pi t)$ Volts. Calculate the instantaneous voltage exactly at $t = \frac{1}{600}\text{ s}$.
Topic 7.2: AC applied to Pure R, L, and C
9.
Using the concepts illustrated in the prompt above, physically explain *why* the current leads the voltage by $90^\circ$ in a pure capacitor. Base your reasoning on the rate of charge accumulation.
AI Image Prompt:
A clean phasor diagram and corresponding wave graph for a purely capacitive AC circuit. Left side: A phasor diagram with V_0 phasor on the X-axis and I_0 phasor pointing straight up along the positive Y-axis, showing a 90-degree lead. Right side: A time-domain wave graph plotting sinusoidal Voltage (V) and Current (I) curves, where the current wave peaks a quarter-cycle before the voltage wave. Use standard physics notation. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q9_PureCapacitorPhasor.jpg
10.
Prove mathematically that the alternating current lags behind the alternating voltage by $\pi/2$ radians in a purely inductive circuit.
11.
An ideal pure inductor of $0.5\text{ H}$ is connected directly to a $200\text{ V}, 50\text{ Hz}$ AC supply. Calculate the peak current flowing through the inductor.
12.
A $100\text{ }\mu\text{F}$ capacitor is connected in series with a $40\text{ V}, 60\text{ Hz}$ AC source. Find the maximum (peak) current in the circuit.
13.
How does the inductive reactance of a coil change if a soft iron core is completely withdrawn from inside it while operating in an AC circuit?
14.
Calculate the RMS current in a purely resistive $50\text{ }\Omega$ circuit if the peak alternating voltage applied is $311\text{ V}$.
15.
What is the exact time lag between the instant of maximum voltage and the instant of maximum current in a pure inductor connected to a $50\text{ Hz}$ AC source?
16.
A pure capacitor blocks DC. Provide a mathematical justification for this statement using the concept of capacitive reactance.
Topic 7.3: Reactance and Impedance
17.
Define impedance. Using the phasor diagram approach described above, derive the standard expression $Z = \sqrt{R^2 + (X_L - X_C)^2}$ for a series LCR circuit.
AI Image Prompt:
A clear schematic of a Series LCR AC circuit connected to an AC voltage source V = V_0 sin(omega*t). Beside it, a fully labeled Phasor Diagram for the case where X_L > X_C. The horizontal phasor is Current (I). The voltage phasors are V_R (horizontal), V_L (pointing up), and V_C (pointing down, shorter than V_L). The resultant voltage V makes an angle phi with the current phasor. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q17_LCRPhasorImpedance.jpg
18.
A coil has a true ohmic resistance of $30\text{ }\Omega$ and an inductive reactance of $40\text{ }\Omega$ at a frequency of $50\text{ Hz}$. If a $200\text{ V}, 100\text{ Hz}$ AC source is applied to it, what will be its new impedance?
19.
A capacitor of $50\text{ }\mu\text{F}$ and a resistor of $10\text{ }\Omega$ are connected in series with a $220\text{ V}, 50\text{ Hz}$ AC source. Find the maximum (peak) current in the circuit.
20.
Calculate the phase angle ($\phi$) by which the current leads or lags the voltage in the specific RC circuit described in Q19. State explicitly if it leads or lags.
21.
A DC electric bulb rated at $60\text{ V}, 10\text{ W}$ is to be operated on a $100\text{ V}, 50\text{ Hz}$ AC mains supply. Calculate the value of the pure inductance required in series to run the bulb safely.
22.
Why do we practically use a choke coil to control alternating current rather than a simple rheostat (variable resistor)? Explain based on power dissipation.
23.
What is the effective equivalent impedance of a series LCR circuit operating exactly at the condition where $X_L = X_C$? What is this specific electrical state called?
24.
A $100\text{ mH}$ inductor, a $20\text{ }\mu\text{F}$ capacitor, and a $10\text{ }\Omega$ resistor are connected in series to an AC source. Calculate the total impedance of the circuit at a frequency of $100\text{ Hz}$.
Topic 7.4: Resonance
25.
Derive the mathematical expression for the resonant frequency ($f_r$) of a series LCR circuit from the fundamental condition of resonance.
AI Image Prompt:
A 2D Cartesian graph plotting Impedance (Z) on the Y-axis against Frequency (f) on the X-axis for a series LCR circuit. The curve is roughly V-shaped or U-shaped. At low frequencies, Z is high (dominated by X_C). As frequency increases, Z drops to a minimum value at the resonant frequency f_r (where Z=R), and then rises again at higher frequencies (dominated by X_L). The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q25_ImpedanceResonance.jpg
26.
Define the Q-factor. Explain theoretically how the ohmic value of $R$ affects the sharpness of the resonance curve.
27.
In a series LCR circuit, $R=20\text{ }\Omega$, $L=1.5\text{ H}$, and $C=35\text{ }\mu\text{F}$. Find the resonant angular frequency ($\omega_r$) and the Q-factor of the circuit.
28.
If the capacitance in a given resonant LCR circuit is suddenly made 4 times its original value, how exactly should the inductance be changed to keep the resonant frequency completely unchanged?
29.
Calculate the bandwidth (the difference between the two half-power frequencies) for the specific circuit described in Q27.
30.
What is the exact phase difference between the instantaneous voltage across the inductor ($V_L$) and the instantaneous voltage across the capacitor ($V_C$) when the LCR circuit is operating at resonance?
31.
A radio receiver can tune over the AM broadcast band from $800\text{ kHz}$ to $1200\text{ kHz}$. If its internal LC tuning circuit has an effective fixed inductance of $200\text{ }\mu\text{H}$, what must be the required range of its variable capacitor?
32.
Explain the phenomenon of "Voltage Magnification." Why does the voltage across the inductor (or capacitor) sometimes wildly exceed the total applied source voltage in a series resonant circuit?
Topic 7.5: Power in AC Circuit
33.
Define Power Factor. Using trigonometric integration over a full cycle, derive the standard power equation $P_{avg} = V_{rms}I_{rms}\cos\phi$.
AI Image Prompt:
A clear Right-Angled Power Triangle. The horizontal base represents True Power (P) in Watts. The vertical side represents Reactive Power (Q) in VAR. The hypotenuse represents Apparent Power (S) in VA. The angle between the base (P) and hypotenuse (S) is labeled phi (φ). Use standard physics notation. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q33_PowerTriangleAdv.jpg
34.
A $220\text{ V}, 50\text{ Hz}$ AC source is connected to a pure inductance of $0.2\text{ H}$ and a pure resistance of $20\text{ }\Omega$ in series. Calculate the power factor of this circuit.
35.
What is meant by the "wattless component" of AC current? Calculate the magnitude of the wattless current if the total RMS current is $10\text{ A}$ and the phase angle is $\phi = 60^\circ$.
36.
A choke coil and an incandescent bulb are connected in series with an AC source, and the bulb shines brightly. How does its brightness change if a soft iron core is slowly inserted into the choke? Give reasoning.
37.
An industrial electric load consumes $1\text{ kW}$ of active power at a lagging power factor of $0.8$ from a $220\text{ V}, 50\text{ Hz}$ mains line. Calculate the value of the pure capacitance that must be connected in parallel to correct the power factor to exactly unity ($1.0$).
38.
Is it physically possible for the power factor ($\cos\phi$) of a passive LCR circuit to be negative? Explain based on phase angles.
39.
A specific LCR circuit has $R=400\text{ }\Omega$, $X_L=200\text{ }\Omega$, and $X_C=200\text{ }\Omega$. What is the power factor of this circuit? State the electrical condition it is in.
40.
Show mathematically by integrating instantaneous power ($P = VI$) that the average active power consumed in a purely capacitive circuit over one complete cycle is exactly zero.
Topic 7.6: LC Oscillations
41.
Using the visual sequence described, qualitatively explain the working of LC oscillations and the continuous transfer of energy between electric and magnetic fields.
AI Image Prompt:
A schematic sequence of 4 panels showing a complete cycle of LC oscillations. Panel 1: Capacitor fully charged, no current. Panel 2: Capacitor discharging, current max through inductor (magnetic field shown). Panel 3: Capacitor oppositely charged, no current. Panel 4: Capacitor discharging again, current max in opposite direction. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q41_LCOscillationCycle.jpg
42.
A $1.5\text{ }\mu\text{F}$ capacitor is charged fully to $60\text{ V}$. The charging battery is then disconnected, and a pure $15\text{ mH}$ coil is connected across it to initiate oscillations. Find the absolute maximum current that will flow in the coil.
43.
Prove mathematically that the total energy ($U = U_E + U_B$) of an ideal, resistanceless LC circuit is strictly conserved during its oscillations at any arbitrary time $t$.
44.
Calculate the oscillation frequency ($f$) for the specific LC circuit established in question Q42.
45.
Draw a direct analogy between the differential equation governing LC electrical oscillations and the equation governing the mechanical oscillations of a simple block-spring system. Identify the corresponding terms.
46.
If the resistance of the inductor in a practical LC circuit is NOT zero, what happens to the oscillations over time? Draw a rough generic graph of charge ($q$) vs time ($t$) to illustrate this.
47.
At what exact time ($t$) after the circuit is connected in Q42 is the total energy of the system shared perfectly equally between the electric field of the capacitor and the magnetic field of the inductor? (Express in terms of time period $T$).
48.
Why cannot LC oscillations be sustained indefinitely in any practical laboratory circuit without coupling it to an external active energy source (like a transistor oscillator)? State two reasons.
Topic 7.7: Transformers
49.
Explain the principle and working of a transformer using Faraday's laws of electromagnetic induction. Derive the transformer ratio equation.
AI Image Prompt:
A clear schematic of a Step-Up Transformer. It features a rectangular laminated soft iron core. The primary winding (left) has few turns of thick wire connected to an AC source V_p. The secondary winding (right) has many turns of thinner wire connected to a load V_s. Magnetic flux lines are shown circulating through the iron core. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level2_Q49_StepUpTransformer.jpg
50.
A step-down transformer converts a high transmission line voltage from $2200\text{ V}$ to $220\text{ V}$. The primary coil has $4000$ turns. If the transformer efficiency is $90\%$ and it delivers an output power of $8\text{ kW}$, find the total input power and the number of turns in the secondary coil.
51.
Calculate the actual primary current ($I_p$) and secondary current ($I_s$) for the non-ideal transformer operating under the conditions given in Q50.
52.
Why is the heavy core of a commercial transformer built up using thin laminated sheets of steel rather than casting it as a single solid block of metal?
53.
Explain briefly the physical causes of the following energy losses in a transformer: (a) Copper loss, (b) Iron (Eddy) loss, (c) Flux leakage, and (d) Magnetostriction (Humming).
54.
A small transformer has $500$ primary turns and $10$ secondary turns. If the primary voltage is $120\text{ V}$ AC, what is the theoretical secondary voltage? Categorize this as a step-up or step-down transformer.
55.
Explain mathematically why a transformer completely fails to change the voltage of a Direct Current (DC) power supply (like a car battery).
56.
Determine the voltage transformation ratio ($k$) and the efficiency ($\eta$) of a transformer if it draws a primary current of $5\text{ A}$ at $200\text{ V}$ and delivers a secondary current of $40\text{ A}$ at $20\text{ V}$.