Kcl Kvl In Ac Circuits

The three elements X, Y, and Z are connected in series across an AC source. From the previous question, we know: - Element X is a resistor, with impedance . - Element Y is an inductor, with impedance , where is the angular frequency and is the inductance. - Element Z is a capacitor, with impedance , where is the capacitance. The total impedance of the series circuit is the sum of the individual impedances: Thus, the impedance of the series circuit is: ### Step 1: Variation of current with frequency The current in the series circuit is given by Ohm's law: Since depends on the frequency , the current will vary with . - When is low, the capacitive reactance is high, so the current is lower. - When increases, the inductive reactance increases, and the capacitive reactance decreases, so the current starts increasing until it reaches a peak. - Beyond a certain frequency, the inductive reactance dominates, and the current decreases. Thus, the current shows a resonant behavior with frequency, with the impedance being lowest at a certain resonant frequency . % Correct Answer Correct Answer:} - Impedance of the series circuit:

Step 1: For element , voltage and current are in phase, which means that the impedance is purely resistive. Therefore, must be a resistor.
Step 2: For element , leads by . This suggests that element behaves like an inductor, because in an inductor, the current lags the voltage, and at high frequencies, the phase difference between voltage and current increases, but here we are given that voltage leads current, which is a characteristic of inductive reactance at a specific frequency.
Step 3: For element , leads by , which indicates capacitive behavior, because in a capacitor, current leads voltage by at high frequencies, and at lower frequencies, this phase difference decreases.
Conclusion:
- is a resistor,
- is an inductor,
- is a capacitor.

The electric field due to a long straight charged wire with a linear charge density at a distance from it is given by the formula:
where:
(charge density of wire AB),
(charge density of wire CD),
(distance from the point to each wire), is the permittivity of free space.
The net electric field at the midpoint between the two wires is the vector sum of the fields due to each wire. Let’s calculate the electric fields and at the midpoint:
Step 1: Calculate the net electric field The direction of the electric fields due to both wires:
The electric field due to wire AB (with ) points away from the wire, toward wire CD.
The electric field due to wire CD (with ) points toward the wire, so it also points toward wire AB.
Since both fields point toward wire CD, the net electric field is the sum of the magnitudes:
Thus, the magnitude of the net electric field is:
and the direction is toward wire CD (left). % Correct Answer Correct Answer:}
The magnitude of the net electric field is . The direction of the net electric field is toward wire CD (left).

Step 1: Understand the impedance of a series LC circuit.
In a series LC circuit, the total impedance is given by: where (inductive reactance) and (capacitive reactance), and is the angular frequency. Step 2: Determine the resonance condition.
Resonance occurs when the inductive and capacitive reactances are equal: Step 3: Analyze the circuit before resonance.
Before resonance, the frequency is less than the resonance frequency , so : Thus, the circuit is capacitive before resonance. Step 4: Analyze the circuit after resonance.
After resonance, the frequency is greater than the resonance frequency , so : Thus, the circuit is inductive after resonance. Step 5: Match with the options.
Before resonance: capacitive; after resonance: inductive. This matches option (C).

Since the element identified in part (a) is an inductor, the inductive reactance is given by: where is the frequency of the applied AC voltage. From the formula, it is evident that the inductive reactance is directly proportional to the frequency . Therefore, as frequency increases, the reactance increases linearly.

The graph is a straight line starting from the origin, confirming that .

In an AC circuit, when the current lags behind the voltage by , the element causing this phase difference is a pure inductor. This is because in an inductor, voltage leads current by 90°, or equivalently, current lags voltage by 90°. Hence, the element is an inductor. The reactance of an inductor is given by: where = inductive reactance, = angular frequency of AC supply, = inductance of the coil.