In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current ...
In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current (in Amperes) through the resistor is 2sin(𝑡 + 𝜋/2). The displacement current (in Amperes) through the capacitor is _________.
Answer: Option C (2sin(𝑡 + 𝜋/2)
Solution:
The displacement current through the capacitorIn a series circuit containing an ideal resistor and an ideal capacitor, the displacement current through the capacitor is equal to the conduction current through the resistor at every instant. No actual charge carriers move through the dielectric between the capacitor plates. Instead, the changing electric field in the dielectric gives rise to the displacement current. For an ideal capacitor, the displacement current through the dielectric is exactly equal to the conduction current in the connecting wires and through the resistor. Consequently, if the conduction current is sinusoidal, the displacement current is also sinusoidal with the same amplitude, frequency, and phase.
| Property | Ideal Resistor (R) | Ideal Capacitor (C) |
|---|---|---|
| Primary Function | Dissipates energy as heat via conduction. | Stores energy; allows current flow via displacement field. |
| Current Type | Conduction Current ($I_c$): Flow of actual charge carriers. | Displacement Current ($I_d$): Arises from a changing electric field. |
| Series Relationship | Current is $2\sin(t + \pi/2)$. | Current is identical: $2\sin(t + \pi/2)$. |
| Continuity Rule | $I_{wire} = I_{resistor}$ | $I_{conduction} = I_{displacement}$ (at every instant). |
| Charge Carriers | Electrons move through the material. | No charge carriers cross the dielectric gap. |
| Physical Mechanism | Collision of electrons with atoms. | Time-varying electric flux ($\frac{d\Phi_E}{dt}$). |
| Voltage–Current Relationship | \(V = IR\) | \(I = C\frac{dV}{dt}\) |
| Phase Difference | $0^\circ$ (Voltage/Current in phase). | Current leads voltage by $90^\circ$. |