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Chapter 7: Alternating Current - Easy / Standard (Level 1)
Topic 7.1: Average and RMS Values
1.
Define the mean (average) value of an alternating current over a half-cycle. Write its relation with the peak current $I_0$.
2.
Derive the mathematical relationship between the Root Mean Square (RMS) current ($I_{rms}$) and the peak current ($I_0$) for a sinusoidal AC.
3.
The peak voltage of a standard $220\text{ V}$ AC supply is approximately $311\text{ V}$. Briefly explain why $220\text{ V}$ AC is considered more dangerous than $220\text{ V}$ DC.
4.
The equation of an alternating current is given by $I = 50 \sin(100\pi t)$. Find its frequency and the RMS value of the current.
5.
Why does a standard moving coil galvanometer or a standard DC ammeter measure exactly zero when connected to an AC circuit?
6.
Prove mathematically that the average value of an alternating voltage $V = V_0 \sin(\omega t)$ over one complete cycle is zero.
7.
The instantaneous current in an AC circuit is $i = 4 \cos(\omega t + \pi/4)$ Amperes. Find its RMS value.
8.
State one primary electrical reason why Alternating Current (AC) is overwhelmingly preferred over Direct Current (DC) for long-distance power transmission.
Topic 7.2: AC applied to Pure R, L, and C
9.
Based on the diagrams described above, write the standard instantaneous equations for Voltage $V$ and Current $I$ in a purely capacitive circuit, establishing their phase relationship.
AI Image Prompt:
Two side-by-side graphs for a purely capacitive AC circuit. On the left: A Phasor Diagram showing the Current phasor (I_0) leading the Voltage phasor (V_0) by exactly 90 degrees (pi/2) counter-clockwise. On the right: A Wave Diagram plotting V and I vs time (or omega*t), showing the current wave peaking exactly one-quarter cycle before the voltage wave. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level1_Q9_CapacitorPhasorWave.jpg
10.
In a purely resistive AC circuit, what is the phase difference between voltage and current? Write their instantaneous equations if $V = V_0 \sin(\omega t)$.
11.
Why does a pure inductor not dissipate any average active power over a complete cycle in an AC circuit? Explain conceptually or mathematically.
12.
An AC source given by $V = 200 \sin(314t)$ Volts is applied across a pure $100\text{ }\Omega$ resistor. Calculate the RMS current flowing through the resistor.
13.
Derive the expression for the instantaneous current in a purely inductive circuit when the applied alternating voltage is $V = V_0 \sin(\omega t)$.
14.
Conceptually explain why a capacitor perfectly blocks steady Direct Current (DC) but allows Alternating Current (AC) to "pass" through the circuit easily.
15.
A $60\text{ }\mu\text{F}$ capacitor is connected directly to a $110\text{ V}, 60\text{ Hz}$ AC supply. Calculate the RMS current in the circuit. (Take $\pi \approx 3.14$).
16.
A pure inductor of $25.0\text{ mH}$ is connected to a $220\text{ V}, 50\text{ Hz}$ AC source. Find the RMS current through the inductor.
Topic 7.3: Reactance and Impedance
17.
Explain the physical reasoning behind the variation of $X_L$ and $X_C$ with frequency $f$, as depicted in the graph above.
AI Image Prompt:
A Cartesian graph displaying Reactance (X) on the Y-axis versus Frequency (f) on the X-axis. Two distinct curves are plotted: 1) A straight line passing through the origin sloping upwards, labeled X_L (Inductive Reactance). 2) A rectangular hyperbola curve in the first quadrant, decreasing asymptotically towards the X-axis, labeled X_C (Capacitive Reactance). The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level1_Q17_ReactanceFrequency.jpg
18.
Define the terms "Impedance" and "Admittance" in an AC circuit. Give their respective SI units.
19.
Draw a clear Impedance Triangle for a series LCR circuit operating under the condition where Inductive Reactance is greater than Capacitive Reactance ($X_L > X_C$). Label $R$, $X_L - X_C$, $Z$, and the phase angle $\phi$.
20.
Define the term "Susceptance". How is it related to reactance, and what is its SI unit?
21.
What is the exact phase difference between the voltage across the inductor ($V_L$) and the voltage across the capacitor ($V_C$) in a series LCR circuit?
22.
A resistor of $200\text{ }\Omega$ and a capacitor of $15.0\text{ }\mu\text{F}$ are connected in series to a $220\text{ V}, 50\text{ Hz}$ AC source. Calculate the total impedance ($Z$) of the circuit.
23.
For the circuit in Q22, calculate the RMS current flowing through the circuit.
24.
How does the total impedance of a standard series LCR circuit behave at extremely high frequencies and at extremely low frequencies?
Topic 7.4: Resonance
25.
State the defining electrical condition for resonance in a series LCR circuit. Using this condition, derive the formula for the resonant frequency ($f_r$).
26.
Based on the resonance curves shown above, explain what the "sharpness" of the curve indicates about the circuit's selectivity and its relationship with resistance $R$.
AI Image Prompt:
A Resonance Curve graph showing RMS Current (I_rms) on the Y-axis versus Angular Frequency (omega) on the X-axis for a series LCR circuit. Plot two bell-shaped curves peaking at the same central frequency (omega_r). One curve is sharp and tall, labeled 'Small R'. The second curve is broad and flat, peaking much lower, labeled 'Large R'. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level1_Q26_ResonanceCurve.jpg
27.
Define the Quality Factor (Q-factor) of a resonant circuit. Write its mathematical formula in terms of $L$, $C$, and $R$.
28.
What happens to the resonant frequency of an LCR circuit if a soft iron core is suddenly inserted into the core of the inductor? Justify your answer.
29.
Define the term "Bandwidth" ($\Delta \omega$) of a resonant LCR circuit. How is it related to the resonant frequency and the Q-factor?
30.
An LCR series circuit has $L = 2.0\text{ H}, C = 32\text{ }\mu\text{F}$ and $R = 10\text{ }\Omega$. Find the resonant angular frequency ($\omega_r$).
31.
Calculate the Quality factor (Q-factor) for the circuit given in Q30.
32.
Why is a series LCR resonant circuit technically referred to as an "acceptor circuit" in radio and communication tuning?
Topic 7.5: Power in AC Circuit
33.
Define Power Factor in an AC circuit. What is its maximum possible value, and under what specific circuit condition is this maximum value achieved?
34.
Derive the mathematical expression for the average power dissipated over a complete cycle in a series LCR circuit: $P_{avg} = V_{rms} I_{rms} \cos\phi$.
35.
What is meant by "wattless current"? In which specific idealized types of circuits does the current become strictly wattless?
36.
To reduce an alternating current in a circuit, a choke coil is overwhelmingly preferred over a standard ohmic rheostat. Explain the fundamental physical reason for this preference.
37.
An AC circuit has an applied voltage $V = 100 \sin(100t)$ Volts and a resulting current $I = 100 \sin(100t + \pi/3)$ mA. Calculate the average power dissipated in the circuit.
38.
Calculate the power factor of a series RL circuit having a resistance $R = 8\text{ }\Omega$ and an inductive reactance $X_L = 6\text{ }\Omega$.
39.
Prove mathematically that the average active power consumed in a purely capacitive circuit over one full cycle is exactly zero.
40.
How can the poor power factor of a highly inductive industrial circuit (like one with many electric motors) be improved or "corrected"?
Topic 7.6: LC Oscillations
41.
Explain qualitatively the underlying physical principle of LC oscillations, focusing on the continuous exchange of energy.
42.
Write the fundamental differential equation governing the charge $q$ in an ideal, resistanceless LC circuit. State the formula for its natural angular frequency ($\omega_0$).
43.
Show mathematically that the total energy ($U_E + U_B$) in an ideal, resistance-free LC oscillating circuit remains perfectly constant at any instant of time $t$.
44.
Compare electrical LC oscillations with the mechanical oscillations of a simple block-spring system. Specifically, what is the exact electrical analogue of mechanical mass ($m$) and the spring constant ($k$)?
45.
A $1.5\text{ }\mu\text{F}$ capacitor is charged to a potential difference of $50\text{ V}$ and then connected across a pure $20\text{ mH}$ inductor to start oscillations. Calculate the maximum current that will flow in the circuit.
46.
Calculate the frequency ($f$) of the electromagnetic waves radiated by an LC antenna circuit consisting of a $1\text{ mH}$ inductor and a $1\text{ nF}$ capacitor.
47.
In a real, practical LC circuit, the oscillations do not continue indefinitely; they are "damped". Provide two primary physical reasons why these oscillations eventually die out.
48.
In an oscillating LC circuit, at the exact instant when the current through the inductor is maximum, what is the value of the charge stored on the capacitor?
Topic 7.7: Transformers
49.
State the underlying principle of a transformer. Can a transformer be used to step up or step down a steady Direct Current (DC) voltage?
50.
Distinguish clearly between a step-up and a step-down transformer strictly on the basis of their turns ratio ($N_s / N_p$) and output voltage.
AI Image Prompt:
A schematic diagram of a Step-Down Transformer. Show a rectangular laminated iron core. On the left arm, draw a Primary Coil with a large number of tight turns connected to an AC source V_p. On the right arm, draw a Secondary Coil with visibly fewer turns connected to a load resistor R_L and labeled V_s. The background of the whole image should be fully white, in landscape mode, mathematically correct, and high quality.
Filename: Level1_Q50_StepDownTransformer.jpg
51.
Mention three major sources of energy loss in a practical transformer and state briefly how each is minimized in commercial designs.
52.
What is meant by "Copper Loss" ($I^2R$ loss) in a transformer? How is this specific loss minimized for the low-voltage, high-current coil?
53.
An ideal transformer is designed to step down a $2200\text{ V}$ primary voltage to a safe $220\text{ V}$ secondary voltage. If the primary coil possesses $2000$ turns, how many turns must be wound on the secondary coil?
54.
In the transformer from Q53, if the secondary is connected to a $22\text{ }\Omega$ resistive load, calculate the currents flowing in the primary and secondary coils. (Assume $100\%$ efficiency).
55.
Why is electrical power generated at power stations transmitted over hundreds of kilometers at extremely high voltages (e.g., $132\text{ kV}$ or $400\text{ kV}$) rather than at standard domestic voltages? Explain using the concept of power loss.
56.
What is the specific purpose of using a "soft iron" core inside a transformer, instead of steel or an air core?