Displacement Current: Definition, Formula and Ampere-Maxwell Law

The change in position of an object is called Displacement. Displacement has a direction and magnitude, being a vector quantity. It is when the object changes its position from one point to another. Now, not just conduction current, there is also a different kind of current named the Displacement current. Just like the conduction current, the displacement current also does not show from the real movement of electric charge.

Key Terms: Current, Ampere, Magnetic field, Law of Induction, Maxwell law’s, Magnetic field lines.

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Basic Idea of Displacement Current

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In Maxwell’s equations, displacement current is a quantity. To explain Displacement current better, it is put in terms of the rate of change of the electric displacement field (D). With the help of a capacitor, the phenomenon observed can be well described. An electric field produces a magnetic field. J.C. Maxwell proposed that for logical consistency. Similarly, an electric field that is changing should also produce a magnetic field. Maxwell proposed this current be referred to as the displacement current and suggested that this current was proportional to the electric field’s rate of change. This displacement current arises due to the presence of the electromotive force that is varying in nature. This can be understood through the process of charging a capacitor. 

The Basic Idea of Displacement Current

When a capacitor is charging, no conduction is to be observed between the plates of the conductor. But, with time due to the change in accumulation of charges above the plates of the conductor, the electric field changes. This leads to displacement current which is of the form –

\(I_D = J_D S = S \frac{\partial D}{\partial t}\)

Where

• I_D refers to the displacement current
• J_D refers to the displacement current density
• S refers to the area of the capacitor plate
• ε is the permittivity of the medium in between the plates
• D is related to the electric field E as D=εE

So, displacement current can be calculated by-

i_d = ε₀ (dQ/dt)

The SI unit for this displacement current is Ampere(A).

The sources of a magnetic field can be-

1. The conduction electric current occurs due to the flow of the charges.
2. The displacement current occurs due to the rate of change of the electric field.

Therefore, the total current, which can be denoted as ‘i’, can be calculated as follows:

i = i_c + i_d

              = i_c + ε₀(dQ/dt)

Where i = the total current

i_c = conduction current

i_d = displacement current

From the above equation, it can be implied that- 

• Outside of the plates of the capacitor: i_c=i and i_d=0
• Inside of the plates of the capacitor: i_c=0 and i_d=i

Hence, it can be said that the current that arises due to the changing electric field is known as the displacement current.

The Ampere’s laws when generalized accordingly can transform into what is known as Ampere-Maxwell Law which is as follows:

“The total current passing through any surface of which the closed-loop is the perimeter is the sum of the conduction current and the displacement current.”

The video below explains this:

Maxwell’s Equations Detailed Video Explanation:

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Ampere-Maxwell Law and Faraday’s Law of Induction

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In accordance with Faraday’s law of induction, there exists an induced emf that is in value equal to the rate of change of the magnetic flux. As emf between any 2 points can be defined as the work done per unit charge to take it from one point to any other. The existence of this emf implies the existence of the electric field. Hence, if Faraday’s law can be rephrased, it can be written as-

A magnetic field that changes with time gives rise to an electric field.

Therefore, it can be concluded that an electric field that changes with time gives rise to a magnetic field as a consequence of the displacement current is the source of the magnetic field that exists. So, it will be fair to say that time-dependent magnetic and electric fields give rise to each other.

Faraday’s law of electromagnetic induction says that a time-varying magnetic field induces an emf, and Maxwell says, an electric field sets up a current and hence a magnetic field. Displacement current is that kind of a current. What follows is that a time-varying electric field produces a magnetic field and vice-versa. Therefore, the electric and magnetic fields have asymmetric behavior. 

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Things to Remember

• Along a closed path at any time, the sum of conduction current and displacement current remains constant.
• At times, where with time, the electric field does not change, as in steady electric fields in a conducting wire, the displacement current might be zero.
• At times, when there is a time-varying electric field and no conduction current, displacement current is present only. Hence, in these situations even when there is no source of conduction current nearby, one can observe a magnetic field present. 

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Previous Year Questions 

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