---
title: "Conduction and displacement current in capacitor"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:conduction-displacement-current-capacitor"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/conduction-displacement-current-capacitor
site: BCT Engineering
---

# Conduction and displacement current in capacitor

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2074 Asoj, 3 marks
- Video: [Displacement current in capacitor | Maxwell's equation | Electromagnetic Waves | Shiva Panchakshari](https://www.youtube.com/watch?v=rcTCVo3jLt0) — Prof. Shiva panchakshari T G
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/conduction-displacement-current-capacitor
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Derive conduction and displacement current in a capacitor.

## Short answer

In a parallel-plate capacitor with v(t)=Vm sin ωt, conduction current in wires is i_c = C dv/dt = C Vm ω cos ωt. Between plates, D = ε v/d, displacement current density J_d = ε/d dv/dt, total i_d = J_d S = C dv/dt = i_c. Thus displacement current equals conduction current, ensuring continuity.

## Answer

Consider a parallel-plate capacitor with plate area S, separation d and dielectric permittivity ε, connected to a source v(t) = V_m sin ωt.

#### Conduction current
The current in the connecting wires is the rate of change of charge on the plates:

```text
i_c = dq/dt = C dv/dt,   C = εS/d
i_c = C V_m ω cos ωt
```

No charge crosses the dielectric, so there is no conduction current between the plates (σ ≈ 0). Without any other current, Ampere's law would give a break in current at the capacitor.

#### Displacement current
Between the plates, E = v/d, so D = εE = εv/d. The displacement current density is

```text
J_d = ∂D/∂t = (ε/d) dv/dt
```

Total displacement current through the plate area:

```text
i_d = J_d S = (εS/d) dv/dt = C dv/dt = C V_m ω cos ωt
```

#### Result
i_d = i_c. The displacement current in the dielectric is equal to the conduction current in the wire, so the total current is continuous through the whole circuit. This is why Maxwell added ∂D/∂t to Ampere's law: ∇×H = J + ∂D/∂t.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a capacitor driven by v = V_m sin wt, what is the conduction current in the connecting wires?
   A. C V_m w cos wt, from i = C dv/dt
   B. C V_m sin wt
   C. V_m / C
   D. C V_m w sin wt
   Answer: A. The wire current is the rate of change of plate charge, i = dq/dt = C dv/dt, which gives C V_m w cos wt. (Key point: Conduction current in wires)

2. Why is there a break in conduction current at the capacitor?
   A. The wires have high resistance
   B. The source frequency is too low
   C. No charge crosses the dielectric between the plates
   D. The plates absorb all the current
   Answer: C. The dielectric has sigma ~ 0, so no free charge moves across it and conduction current cannot flow between the plates. (Key point: No conduction between plates)

3. What is the displacement current i_d between the plates compared with i_c in the wire?
   A. i_d is zero
   B. i_d = i_c, so total current is continuous
   C. i_d is larger than i_c by the factor d
   D. i_d is 90 degrees out of phase and half the size
   Answer: B. i_d = J_d S = (eS/d) dv/dt = C dv/dt, which is exactly i_c, so the current is continuous around the circuit. (Key point: Displacement current equals conduction current)

## Related in Electromagnetism

- [Displacement vs conduction current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-vs-conduction-current) — Electromagnetism, chapter 3, asked 2×
- [Frequency where conduction equals displacement](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-conduction-ratio-soil) — Electromagnetism, chapter 3
- [Displacement current density at 1 kHz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-metal-1khz) — Electromagnetism, chapter 3
- [Maxwell's equations, displacement current](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-explain-displacement-depth) — Electromagnetism, chapter 3
- [Displacement current density in transformer](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-density-transformer) — Electromagnetism, chapter 3
- [Displacement current density at 60 Hz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adisplacement-current-metallic-conductor-60hz) — Electromagnetism, chapter 3

## Related in other subjects

- [Displacement current in parallel plate capacitor](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-circular-plates-b-integral) — Engineering Physics, chapter 9
- [Induced magnetic field, displacement current](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-induced-field-circular-plates) — Engineering Physics, chapter 9
- [Displacement current density, dE/dt, induced B](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-density-dedt-plates) — Engineering Physics, chapter 9
- [Displacement current: B at 50 mm](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-density-field-fifty-mm) — Engineering Physics, chapter 9