Displacement Current

Displacement current is an important idea in electromagnetism, introduced by physicist James Clerk Maxwell. He came up with this concept to explain how a changing electric field can create a magnetic field—even when there's no actual electric current flowing. This was a major breakthrough that helped form the foundation of Maxwell’s equations, which link electricity and magnetism into one unified theory. Displacement current is especially useful for explaining how electromagnetic waves (like light and radio signals) can move through empty space or insulating materials where no physical current exists. In this blog, we’ll explain what displacement current really means, why it matters, and how it affects the technology we use every day.

1.0Definition of Displacement Current

Displacement current refers to the current that emerges alongside conduction current when there is a time-varying electric field, leading to a change in electric flux.

$I_D={ϵ}_0\frac{d{\varphi }_E}{dt}$

2.0The Inconsistency in Ampere’s Circuital Law

• According to Ampere’s Circuital Law, the line integral of the magnetic field $\vec{B}$ around any closed path C is directly proportional to the net electric current Ienclosed by the loop. $\oint \vec{B}.\overrightarrow{dl}={\mu }_{0}I$…………(1)
• Maxwell demonstrated that the above equation was logically incomplete and required modification to remain consistent in all situations.
• To demonstrate this inconsistency, consider a parallel plate capacitor being charged by a battery, as shown in the circuit diagram. During the charging process, a time-varying current I flows through the connecting wires. This changing current generates a magnetic field in the region around the capacitor.
• Consider two planar loops, $C_1$ and $C_2$ ​: loop $C_1$​ is positioned just to the left of the capacitor, while loop $C_2$ lies between the plates of the capacitor, with both loops oriented such that their planes are parallel to the plates.
• The current I  passes through the area enclosed by loop $C_1$​, since the connecting wire carrying the current intersects this loop. According to Ampère’s Law, this current contributes to the magnetic field around $C_1$. ${\oint }_{c_1}\vec{B}.\overrightarrow{dl}={\mu }_{0}l$..... (2)
• However, the area enclosed by loop $C_2$ lies entirely between the plates of the capacitor, where no conduction current flows through.  ${\oint }_{c_2}\vec{B}.\overrightarrow{dl}=0$…………….(3)
• Now, imagine the loops $C_{1}and{C}_2$​ positioned infinitesimally close to each other. In that case, the line integrals of the magnetic field around both loops should be equal: ${\oint }_{c_1}\vec{B}.\overrightarrow{dl}={\oint }_{c_2}\vec{B}.\overrightarrow{dl}$
• This contradiction with equations (2) and (3) revealed the inconsistency, leading Maxwell to recognize the necessity of modifying Ampere’s Law.

Note: Current flows through the wires to the plates of a capacitor, but no real current flows through the gap between the plates. Still, a magnetic field is created in that gap which seems to go against Ampère’s original law.

3.0Displacement Current in Charging Capacitor

To understand how a changing electric field generates a magnetic field, let us examine the process of charging a capacitor.

Case-1

Question: Calculate Magnetic Field at P

Solution: For this, we consider a plane circular loop of radius r whose plane is perpendicular to the direction of the current- carrying wire, and which is centred symmetrically with respect to the wire

Using Ampere’s Law

$\oint \mathbf{B}\cdot d\mathbf{l}={\mu }_0{I}_{\text{enc}}$

$B_p(2\pi r)={\mu }_{0}I$

$B_p=\frac{{\mu }_{0}I}{2\pi r}$

Case-2

Question: Calculate Magnetic Field at P

Solution:Here we consider a different surface, which has the same boundary. This is a pot-like surface which nowhere touches the current, but has its bottom between the capacitor plates.

Using Ampere’s Law

$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$

$B_p(2\pi r)={\mu }_0(0)(\because \text{No current passes through the surface})$

$B_p=0$

So, we have a contradiction; calculated one way, there is a magnetic field at a point P; calculated another way, the magnetic field at P is zero. This shows the inconsistency in ampere circuital law. But what is the reason for this inconsistency? We can easily solve this problem by considering the following situation.

Electric field between the plates

$E=\frac{q}{A{\epsilon }_0}$

Electric flux through shaded area

${\Phi }_E=EA=\frac{q}{A{\epsilon }_0}A$

${\Phi }_E=\frac{q}{{\epsilon }_0}$

Since charge q is varying w.r.t. Time,

$\frac{d{\Phi }_E}{dt}=\frac{1}{{\epsilon }_0}\frac{dq}{dt}$

$\frac{dq}{dt}={\epsilon }_0\frac{d{\Phi }_E}{dt}{i}_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$

Maxwell defined $i_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$ as displacement current

So, this is the solution of inconsistency that we observed. So, in Case2

Using Ampere’s Law

$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{\text{enc}}$

$B_p(2\pi r)={\mu }_0(i{)}_d(\because i_d\text{passes through the surface})$

$B_p=\frac{{\mu }_0{i}_d}{2\pi R}$

The source of a magnetic field is not just the conduction electric current due to flowing charges, but also the time rate of change of electric field.

Hence, total current is defined as $I=I_C+I_D$

So, Ampere’s Circuital Law can be written as

$\oint \vec{B}.\overrightarrow{dl}={\mu }_0(I_C+I_D)$ [Also known as Ampere Maxwell's Law]

Note: Displacement current also solved the current continuity problem

4.0Properties of Displacement Current

1. Occurs with Changing Electric Flux: Displacement current arises when the electric flux changes; it doesn't exist under steady conditions.
2. Not a Real Current: It’s not an actual flow of charge but contributes to the magnetic field in Ampere’s law, hence treated like a current.
3. Equal to Rate of Charge Displacement: Its magnitude equals the rate at which charge moves between capacitor plates.
4. Ensures Continuity: Along with conduction current, it maintains the continuity of current in circuits.