Derive Continuity Equation (Appeared: May 2022, May 2023, Dec 2023)

Continuity Equation (Appeared: May 2022, May 2023, Dec 2023)

The Continuity Equation is a mathematical expression that describes the conservation of electric charge in a semiconductor or conducting medium. It ensures that charge doesn't magically appear or disappear—what goes in must come out unless it is stored or lost.

1. Concept Overview

In telecommunication circuits and semiconductor devices like diodes and transistors:

• Electrons and holes are moving.

• Their movement creates current.

• The density of charge in a region may change with time.

This change is captured by the continuity equation.

2. General Statement

The Continuity Equation is:
∂ρ/∂t + ∇ · J = 0

Where:

• ρ = charge density (C/m³)

• ∂ρ/∂t = rate of change of charge density with time

• ∇ · J = divergence of current density J (A/m²)

• J = current density vector

3. Physical Meaning

• If more current leaves a region than enters, charge inside decreases → ∂ρ/∂t becomes negative.

• If charge builds up in a region, that means not all current is leaving.

In simple words:
Rate of charge increase = Current coming in − Current going out

4. Application in Semiconductors

In semiconductors (e.g. PN junctions):

• Two types of charge carriers: electrons (n) and holes (p).

• Each has its own current:

  • Jn → electron current density

  • Jp → hole current density

We write two continuity equations:

For Electrons:
∂n/∂t = (1/q) ∇ · Jn + G - R

For Holes:
∂p/∂t = -(1/q) ∇ · Jp + G - R

Where:

• n and p = electron and hole concentration

• q = magnitude of electron charge

• G = generation rate (creating new carriers)

• R = recombination rate (carriers disappearing)

5. Derivation Step-by-Step { Write only steps mentioned in bold for 5marks questions}:

Step 1: Charge in a Volume
Let Q be the total charge inside volume V.
Q = ∫_V ρ dV

Step 2: Change in Charge with Time
Rate of change:
dQ/dt = ∫_V ∂ρ/∂t dV

Step 3: Net Current Flow (Outward)
Using Gauss's Divergence Theorem:
I_out = ∮_S J · dA = ∫_V ∇ · J dV

So, total current leaving volume = divergence of current inside volume.

Step 4: Apply Conservation of Charge
Charge can't vanish, so:
dQ/dt = – I_out

This gives:
∫_V ∂ρ/∂t dV = – ∫_V ∇ · J dV

Since this must be true for any volume, we equate the integrands:
∂ρ/∂t + ∇ · J = 0

6. Significance in Electronics and Telecommunication

• Used in MOSFETs, diodes, BJT, and IC design.

• Forms the base for device simulations (e.g. in SPICE).

• Ensures signal integrity in high-speed digital/analogue circuits.

• Useful in electromagnetic wave propagation, transmission lines, and antenna theory.

7. Key Takeaways

• It is a conservation law for electric charge.

• Helps model how charge carriers behave inside a material.

• Without this, telecommunication devices wouldn't function predictably.

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