Displacement Current: Definition, Formula, & Derivation

Displacement current plays a crucial role in electromagnetism as introduced by Maxwell, bridging the gap between conduction current and changing electric fields. Understanding displacement current, its formula, derivation, and distinction from conduction current is essential for mastering topics like Maxwell’s equations and AC circuits. Dive into this page for clear explanations, derivations, formulas, and practical uses of displacement current in physics.

What is Displacement Current? (Definition & Concept)

Displacement current refers to a quantity that extends the concept of electric current to include changing electric fields, especially in situations where no actual charge flows, such as within a capacitor's dielectric. The displacement current was proposed by James Clerk Maxwell to resolve inconsistencies in Ampère's law and unify the laws of electricity and magnetism. In essence, it makes electromagnetic theory self-consistent by introducing a term that acts like a current in the presence of a varying electric field.

Displacement Current Definition (Class 12): Displacement current is the current that arises due to a time-varying electric field and is given by the rate of change of electric displacement field with respect to time.

For example, in a charging capacitor, although no conduction current physically flows through the insulator (dielectric), the changing electric field produces a magnetic field as if a current exists. This is displacement current in action.

Displacement Current Explained Through Maxwell’s Equations

Maxwell identified that Ampère’s law needed modification to account for situations involving changing electric fields but no charge flow, like in capacitors. He introduced the concept of displacement current, denoted by $I_d$, to update Ampère’s law. This refinement led to the famous Maxwell’s equations which accurately describe electromagnetic phenomena.

Displacement current is crucial for explaining electromagnetic wave propagation and forms the theoretical link enabling light to be understood as an electromagnetic wave. This is why in textbooks and competitive exams, “displacement current class 12” is a standard topic, and mastering it is key for deeper physics understanding.

Displacement Current Formula and Key Equations

The formula for displacement current, especially relevant for a parallel plate capacitor or any region with changing electric field, is:

Displacement Current Formula: $I_d = \varepsilon_0 \frac{d\Phi_E}{dt}$

Where:

• $I_d$ = Displacement current (in amperes)
• $\varepsilon_0$ = Permittivity of free space ($8.85\times10^{-12}$ F/m)
• $\Phi_E$ = Electric flux

Alternatively, in terms of the electric field $E$ and area $A$:

Displacement Current Density Formula: The displacement current density, $J_d$, expresses the displacement current per unit area:

Here, $J_d$ is in amperes per square meter $(A/m^2)$, and $\frac{\partial E}{\partial t}$ is the time rate of change of electric field.

Displacement Current Dimensional Formula: $[I_d] = [A]$ (since it has units of ampere, like conduction current).

Displacement Current vs Conduction Current

Understanding the difference between displacement current and conduction current is foundational in physics and electrical engineering:

• Conduction Current ($I_c$): Actual flow of free charges (electrons or ions) through a conductor, given by $I_c = nqAv_d$, where $n$ is charge carrier density, $q$ is charge, $A$ is cross-sectional area, and $v_d$ is drift velocity.
• Displacement Current ($I_d$): Not associated with actual charge movement, but with a changing electric field, and is significant in insulators or during the charging/discharging of capacitors.

Derivation of Displacement Current Equation

The displacement current derivation starts from recognizing a flaw in the original Ampère’s law when applied to a charging capacitor. There, conduction current exists in the wires but not between the plates, yet a magnetic field is observed between the plates. Maxwell’s correction involved introducing displacement current.

1. Ampère’s law (without Maxwell’s correction): $\nabla \times \vec{B} = \mu_0 \vec{J}$
2. Applying to a region inside a capacitor: No conduction current ($J = 0$), but magnetic effects persist.
3. Maxwell introduces displacement current: $\vec{J}_d = \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$
4. Modify Ampère’s law: $\nabla \times \vec{B} = \mu_0 (\vec{J}_c + \vec{J}_d)$
5. So, for a capacitor of area $A$ and electric field $E$: $I_d = \varepsilon_0 A \frac{dE}{dt}$

This step-by-step derivation ensures that magnetic fields are consistently explained whether real charges move (conduction current) or electric fields change (displacement current).

Displacement Current in a Capacitor

Consider an AC circuit with a parallel plate capacitor. When an alternating voltage is applied, the electric field between the plates changes continuously. Although no charge flows through the dielectric, the changing field gives rise to a displacement current.

• Displacement current in a charging capacitor ensures that the total current (conduction + displacement) is continuous throughout the circuit.
• It maintains the magnetic field observed between the capacitor plates, as explained in Maxwell displacement current theory.

For deeper understanding of wave propagation and electromagnetic theory, refer to concepts like wavefronts and Faraday's Law.

Numerical Example: Calculating Displacement Current

Suppose the electric field between a capacitor’s plates of area $0.01\, m^2$ changes at the rate of $2 \times 10^{12}\, V\, m^{-1}\, s^{-1}$. Find the displacement current.

1. Use the displacement current formula: $I_d = \varepsilon_0 A \frac{dE}{dt}$
2. Plug in values: $I_d = (8.85 \times 10^{-12}) \times 0.01 \times (2 \times 10^{12})$
3. $I_d = (8.85 \times 10^{-14}) \times (2 \times 10^{12})$
4. $I_d = 1.77 \times 10^{-1} = 0.177\, A$

This example highlights how to use the displacement current equation and reinforces the concept for exams and practical applications.

Summary Table: Displacement Current vs Conduction Current

The table helps clarify distinctions and similarities. Both types of current generate magnetic fields, which is central to electromagnetic theory.

Displacement Current: Key Points for Exams and Applications

• Makes Maxwell’s equations consistent across all scenarios.
• Enables explanation of electromagnetic waves, light propagation, and AC circuits.
• Fundamental for understanding concepts like electric field due to charges and electromagnetic induction.
• Frequently appears in questions on “displacement current class 12” boards and competitive exams.

To further explore related physics principles, check out topics like oscillatory motion and essential class 12 formulas.

Conclusion

Displacement current is an indispensable concept in electromagnetism, enabling the unification of magnetism and electricity through Maxwell's equations. Its formula, derivation, and distinction from conduction current should be clear to every physics student. For more examples or detailed theory, visit the related displacement current page and keep exploring foundational physics ideas.