Displacement Current — Explained

On this page

1. Conceptual Foundation: The Charging Capacitor Problem
2. Maxwell's Correction and Key Principles
3. Derivation for a Charging Capacitor
4. Real-World Applications and Significance
5. Common Misconceptions
6. NEET-Specific Angle

Detailed Explanation

The concept of displacement current is a profound insight by James Clerk Maxwell that revolutionized our understanding of electromagnetism. Before Maxwell, Ampere's circuital law was a cornerstone of electromagnetism, stating that the line integral of the magnetic field $\vec{B}$ around any closed loop is proportional to the total conduction current $I_c$ passing through the surface bounded by the loop:

$$\oint \vec{B} \cdot dvec{l} = \mu_0 I_c$$

This law worked perfectly for steady currents.

However, Maxwell identified a critical inconsistency when dealing with time-varying electric fields, particularly in the context of a charging capacitor.

Conceptual Foundation: The Charging Capacitor Problem

Consider a parallel plate capacitor being charged by a battery. A conduction current $I_c$ flows through the connecting wires, bringing charge to one plate and removing it from the other. Between the plates, however, there is a vacuum or a dielectric material, and no charge physically flows across the gap. If we apply Ampere's law, we encounter a problem.

Let's choose an Amperian loop $L$ around one of the connecting wires. If we choose a flat surface $S_1$ (like a disc) bounded by $L$ that cuts through the wire, the conduction current $I_c$ passes through $S_1$.

Ampere's law correctly gives a magnetic field around the wire. Now, consider another surface $S_2$ (like a balloon-shaped surface) bounded by the same loop $L$, but this surface passes between the capacitor plates.

Through $S_2$, there is no conduction current ($I_c = 0$) because no charges are flowing across the gap. According to the original Ampere's law, the magnetic field around $L$ should be zero if we use surface $S_2$.

This contradicts the result obtained using $S_1$ and also contradicts experimental observations, which show a magnetic field existing in the region between the plates.

This inconsistency arises because the original Ampere's law implicitly assumes that the current passing through any surface bounded by the loop is the same, which is true only for steady currents (where charge is conserved and continuous).

For time-varying fields, especially when charge accumulates or depletes, this assumption breaks down. The problem is fundamentally related to the continuity equation for charge, which states that $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$, where $\vec{J}$ is current density and $\rho$ is charge density.

Taking the divergence of Ampere's law ($\nabla \times \vec{B} = \mu_0 \vec{J}$), we get $\nabla \cdot (\nabla \times \vec{B}) = \mu_0 \nabla \cdot \vec{J}$. Since the divergence of a curl is always zero, we have $0 = \mu_0 \nabla \cdot \vec{J}$.

This implies $\nabla \cdot \vec{J} = 0$, meaning current density is always divergence-free, which is only true for steady currents. For time-varying currents, $\nabla \cdot \vec{J}$ is not zero, leading to a contradiction.

Maxwell's Correction and Key Principles

Maxwell realized that the changing electric field between the capacitor plates must be responsible for the magnetic field observed there. He looked at Gauss's law for electricity, $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$, or in integral form, $\oint \vec{E} \cdot dvec{A} = \frac{Q_{enclosed}}{\epsilon_0}$.

For the charging capacitor, the charge $Q$ on the plates is changing with time, so $\frac{dQ}{dt} = I_c$. Differentiating Gauss's law with respect to time:

$$\frac{d}{dt} \oint \vec{E} \cdot dvec{A} = \frac{1}{\epsilon_0} \frac{dQ_{enclosed}}{dt}$$

$$\oint \frac{\partial \vec{E}}{\partial t} \cdot dvec{A} = \frac{I_c}{\epsilon_0}$$

This implies that $I_c = \epsilon_0 \oint \frac{\partial \vec{E}}{\partial t} \cdot dvec{A}$.

Maxwell proposed that a term proportional to the rate of change of electric flux should be added to Ampere's law. He defined the displacement current $I_d$ as:

$$I_d = \epsilon_0 \frac{dPhi_E}{dt}$$

where $\Phi_E = \int \vec{E} \cdot dvec{A}$ is the electric flux.

In differential form, the displacement current density $\vec{J}_d$ is given by $\vec{J}_d = \epsilon_0 \frac{\partial \vec{E}}{\partial t}$.

By adding this term, Ampere's law was modified to become the Ampere-Maxwell law:

$$\oint \vec{B} \cdot dvec{l} = \mu_0 (I_c + I_d)$$

$$\oint \vec{B} \cdot dvec{l} = \mu_0 I_c + \mu_0 \epsilon_0 \frac{dPhi_E}{dt}$$

In differential form, this is:

$$\nabla \times \vec{B} = \mu_0 \vec{J}_c + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$$

Now, if we take the divergence of this modified equation:

$$\nabla \cdot (\nabla \times \vec{B}) = \mu_0 \nabla \cdot \vec{J}_c + \mu_0 \epsilon_0 \nabla \cdot \left(\frac{\partial \vec{E}}{\partial t}\right)$$

$$0 = \mu_0 \nabla \cdot \vec{J}_c + \mu_0 \epsilon_0 \frac{partial}{\partial t} (\nabla \cdot \vec{E})$$

Substituting Gauss's law, $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$:

$$0 = \mu_0 \nabla \cdot \vec{J}_c + \mu_0 \epsilon_0 \frac{partial}{\partial t} \left(\frac{\rho}{\epsilon_0}\right)$$

$$0 = \mu_0 \nabla \cdot \vec{J}_c + \mu_0 \frac{\partial \rho}{\partial t}$$

$$\nabla \cdot \vec{J}_c = -\frac{\partial \rho}{\partial t}$$

This is precisely the continuity equation for charge, which expresses the conservation of charge.

Thus, the introduction of displacement current makes Ampere's law consistent with charge conservation.

Derivation for a Charging Capacitor

For a parallel plate capacitor with plate area $A$ and separation $d$, the electric field between the plates is $E = \frac{Q}{\epsilon_0 A}$ (ignoring fringe effects). The electric flux through a surface between the plates (of area $A$) is $\Phi_E = E \cdot A = \frac{Q}{\epsilon_0 A} \cdot A = \frac{Q}{\epsilon_0}$.

The displacement current $I_d$ is then:

$$I_d = \epsilon_0 \frac{dPhi_E}{dt} = \epsilon_0 \frac{d}{dt} \left(\frac{Q}{\epsilon_0}\right) = \frac{dQ}{dt}$$

Since $\frac{dQ}{dt}$ is the rate at which charge flows onto the capacitor plate, it is equal to the conduction current $I_c$ in the wires.

Therefore, in the space between the capacitor plates, $I_d = I_c$. This means that the total current (conduction + displacement) is continuous throughout the circuit, even across the capacitor gap. The conduction current $I_c$ flows in the wires, and the displacement current $I_d$ 'flows' in the gap, ensuring continuity of the total current and thus the magnetic field.

Real-World Applications and Significance

1
Electromagnetic Wave Propagation — The most profound consequence of displacement current is the prediction of electromagnetic waves. Maxwell's equations, with the displacement current term, showed that a time-varying electric field generates a magnetic field, and a time-varying magnetic field (Faraday's law) generates an electric field. This self-sustaining interplay of changing electric and magnetic fields propagating through space constitutes an electromagnetic wave. Without displacement current, EM waves would not be possible in vacuum.
2
Capacitor Behavior — Displacement current explains why capacitors can 'pass' AC current. While no charge physically crosses the dielectric, the changing electric field within the capacitor constitutes a displacement current, which in turn generates a magnetic field, completing the circuit for AC signals. For DC, once the capacitor is fully charged, the electric field becomes constant, $\frac{dPhi_E}{dt} = 0$, and thus $I_d = 0$, effectively blocking DC current.
3
High-Frequency Circuits — At very high frequencies, the rate of change of electric fields becomes significant, and displacement current effects become more pronounced, influencing circuit design and behavior.

Common Misconceptions

Is it a 'real' current? — No, displacement current is not a flow of charge carriers. It does not involve the movement of electrons or ions. It's a conceptual current, a mathematical term that represents the magnetic effect of a changing electric field. It doesn't dissipate energy as heat like conduction current does (unless the dielectric material itself has losses).
Does it exist only in capacitors? — While the charging capacitor is the classic example, displacement current exists wherever there is a changing electric field. This includes propagating electromagnetic waves in free space, where there are no charge carriers at all.
Conduction current vs. Displacement current — Conduction current is the flow of actual charges. Displacement current is due to the change in electric flux. They are fundamentally different phenomena but have the same magnetic effect.

NEET-Specific Angle

For NEET aspirants, understanding displacement current is crucial for several reasons:

Conceptual Clarity — Questions often test the fundamental understanding of what displacement current is, why it was introduced, and its distinction from conduction current.
Maxwell's Equations — Displacement current is a key component of the Ampere-Maxwell law, one of the four Maxwell's equations. Knowledge of these equations is fundamental to electromagnetism.
Electromagnetic Waves — The existence and properties of EM waves are directly linked to displacement current. Questions on EM wave propagation, speed, and nature often implicitly rely on this concept.
Calculations — Numerical problems might involve calculating displacement current given the rate of change of electric flux or electric field, especially in a capacitor. Remember the formula $I_d = \epsilon_0 \frac{dPhi_E}{dt}$ and its relation to $I_c$ in a charging capacitor ($I_d = I_c$).
Units and Dimensions — Be prepared for questions on the units and dimensions of displacement current, which are the same as conduction current (Amperes).

Mastering displacement current means not just memorizing the formula, but truly grasping its conceptual role in completing electromagnetic theory and enabling the existence of light itself.

Often confused with

Side-by-side differences the NEET paper likes to test.

Displacement Current vs Conduction Current

Aspect	Displacement Current	Conduction Current
Nature	Conceptual current, magnetic effect of changing electric field.	Actual flow of charge carriers (e.g., electrons).
Physical Movement of Charge	No physical movement of charge carriers.	Involves physical movement of charge carriers.
Medium	Exists in dielectric media or vacuum.	Exists in conductors.
Energy Dissipation	Does not directly cause Joule heating (energy dissipation).	Causes Joule heating (energy dissipation) due to resistance.
Source	Time-varying electric field (changing electric flux).	Potential difference across a conductor.
Continuity in Capacitor Circuit	Fills the gap between capacitor plates, ensuring continuity of total current.	Flows in connecting wires, stops at capacitor plates.

Displacement current is a theoretical construct representing the magnetic field generated by a changing electric field, without any actual charge movement. It exists in insulators and vacuum. Conduction current, conversely, is the physical flow of charge carriers through a conductor, driven by an electric field, and directly causes energy dissipation as heat.

While both produce magnetic fields, their underlying mechanisms and physical manifestations are fundamentally distinct. Displacement current was crucial for Maxwell to complete the theory of electromagnetism and predict electromagnetic waves.

Why it is tested: For NEET, understanding the distinction between conduction and displacement current is vital for conceptual questions. Students must grasp that displacement current is not a flow of charge but a magnetic effect of a changing electric field, particularly in the context of capacitors and electromagnetic wave propagation. This differentiation helps clarify the role of each in Maxwell's equations.

Questions students ask

5 answered on this topic.

What is the fundamental difference between conduction current and displacement current?

Conduction current is the actual flow of charge carriers (like electrons in a wire) through a material. It involves the physical movement of charge. Displacement current, on the other hand, is not a flow of charge.

It is a conceptual current introduced by Maxwell, representing the magnetic effect produced by a time-varying electric field or, equivalently, a changing electric flux. While both produce magnetic fields, conduction current involves charge transport and energy dissipation (Joule heating), whereas displacement current does not involve charge transport and typically does not dissipate energy.

Why was the concept of displacement current necessary?

The concept of displacement current was necessary to resolve an inconsistency in Ampere's original circuital law when applied to time-varying electric fields, such as those in a charging capacitor. Without it, Ampere's law would violate the principle of charge conservation, as the current would appear discontinuous across a capacitor gap.

Maxwell's addition of the displacement current term made Ampere's law consistent with the continuity equation for charge and, crucially, led to the prediction of electromagnetic waves.

Does displacement current produce heat like a regular current?

No, displacement current does not produce heat (Joule heating) in the same way that conduction current does. Conduction current involves the movement of charge carriers that collide with atoms in the material, dissipating energy as heat.

Displacement current is associated with a changing electric field in a dielectric or vacuum, not with the physical movement of charges. While a dielectric material might have some energy losses when subjected to a changing electric field, this is due to dielectric losses, not directly due to the 'flow' of displacement current itself.

Where does displacement current exist?

Displacement current exists wherever there is a time-varying electric field. The most common example is in the space between the plates of a charging or discharging capacitor, where the electric field is changing. It also exists in free space or any medium where electromagnetic waves are propagating, as EM waves consist of mutually regenerating time-varying electric and magnetic fields. Essentially, any region experiencing a change in electric flux will have an associated displacement current.

What is the Ampere-Maxwell law?

The Ampere-Maxwell law is the corrected form of Ampere's circuital law, incorporating Maxwell's displacement current. It states that the line integral of the magnetic field around any closed loop is proportional to the sum of the conduction current and the displacement current passing through the surface bounded by the loop.

Mathematically, it is expressed as $\oint \vec{B} \cdot dvec{l} = \mu_0 (I_c + I_d)$, or in differential form, $\nabla \times \vec{B} = \mu_0 \vec{J}_c + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$.

This law is one of the four fundamental Maxwell's equations and is crucial for understanding electromagnetic phenomena.