Displacement Current and Maxwell’s Equations: Definition, Derivation of Expression

Displacement Current and Maxwell’s Equations: Maxwell’s equations are a set of equations involving electric field, magnetic field, their sources, the charge and current densities. These equations are known as Maxwell’s equations. Along with the Lorentz force formula, they mathematically express all the basic laws of electromagnetism. The main prediction to emerge from Maxwell’s equations is the existence of E.M. waves, which are time-varying electric and magnetic fields that propagate in space.

The speed of the light waves in vacuum, according to these equations, turned out to be very close to the speed of light \(\left(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) obtained from optical measurements. This explanation led to the remarkable conclusion that light is an electromagnetic (E.M.) wave. Maxwell’s work led to the unification of the domain of electricity, magnetism and light. Heinrich Hertz, in \(1885\), experimentally demonstrated the existence of electromagnetic waves (transverse waves). It was technologically used by Marconi and others, which led to the revolution in communication that we are witnessing today. The electromagnetic spectrum stretching from γ rays (wavelength \(\left.\sim 10^{-12} \mathrm{~m}\right)\) to long radio waves (wavelength \(\left.\sim 10^{6} \mathrm{~m}\right)\) is thus described.

The phenomenon of induction of the magnetic field due to change in the electric field is explained in electromagnetic theory. The magnetic field is produced in the surroundings of the electric current, and it is called conduction current. The concept of displacement current depends on the variation of the Electric field with time. It was developed by the British physicist James Clerk Maxwell in the \(19\)th century. Let’s discuss the displacement current formula and Maxwell’s equations in this article.

Statement of Ampere’s Circuital Law

“The line integral of the magnetic field \(\overrightarrow B \) around any closed curve is equal to \(\mu_{0}\) times the net current \(i\) threading through the area enclosed by the curve.”

i.e \(\oint {\overrightarrow B d\overrightarrow l }  = {\mu _0}\sum i  = {\mu _0}\left( {{i_1} + {i_3} – {i_2}} \right)\)

Where,

\(\mu_{0}=\) permeability of space

\(\overrightarrow B  = \) magnetic field at a point on the boundary of the surface making an angle \({ }^{\prime} \theta^{\prime}\) with the length element ‘\(d\overrightarrow l \)’

\(\oint {\overrightarrow B } d\overrightarrow l  = \) The sum of all the \(\overrightarrow B d\overrightarrow l  = \) products over the complete loop

Note:

1. The total current within the loop in the above figure is \(\left(i_{1}+i_{3}-i_{2}\right)\) . Any current outside this loop is not included in net current, but while calculating \(\oint {\vec B} \cdot \overrightarrow {dl} ,\) we have to include a magnetic field due to all currents (both inside and outside the loop currents).

2. Following is the sign convention to be used: (Outward current ◉ → positive, Inward current ⊗ → negative).

3. When the direction of the current is away from the observer, the closed path’s direction is clockwise and vice versa.

4. This law is valid only for steady or constant currents. This law is valid irrespective of the size and shape of the closed path (Amperian loop) enclosing the current.

5. The statement \(\oint {\overrightarrow B d\overrightarrow l  = 0} \)  does not necessarily mean that the magnetic field \({\overrightarrow B }\) is zero everywhere along the path but that no net current is passing through the path.

Alternative Form of Ampere’s Circuital Law

We know that:

\(\oint {\overrightarrow B d\overrightarrow l  = {\mu _0}\sum {i = {\mu _0}\left( {{i_1} + {i_3} – {i_2}} \right)} } \)

By using \(\mathop B\limits^ \to = {\mu _o}\mathop H\limits^ \to \) (where \(\overrightarrow H\) = magnetizing field)

\(\oint {Bdl = } {\mu _o}\sum {i = } {\mu _o}\left( {{i_1} + {i_3} + {i_2}} \right) = \oint {{\mu _o}\mathop H\limits^ \to \mathop {dl}\limits^ \to } = {\mu _o}\sum i \)

\( \Rightarrow \oint {\mathop H\limits^ \to \mathop {dl}\limits^ \to = \sum i } \)

Inconsistency in Ampere’s Circuital Law

James Clerk Maxwell explained that Ampere’s Law is valid only for steady(constant) current or when the electric field is invariable with time. To study this inconsistency, let us consider a parallel plate capacitor being charged by a battery. During charging, time-varying current flows through connecting wires.

Applying Ampere’s Law for loop \({I_1}\) and \({I_2}\),

For loop \(1\) –  \(\oint {\mathop B\limits^ \to \mathop {dl}\limits^ \to } = {\mu _o}i\)

For loop \(2\) – \(\oint {\mathop B\limits^ \to \mathop {dl}\limits^ \to } = 0\)  ( \(= 0\) between the plates).

But practically, it is observed that there is a magnetic field between the plates when the plate is getting charged or discharged. Hence Ampere’s Law fails:

i.e.  \(\oint {\mathop B\limits^ \to \mathop {dl}\limits^ \to } \ne {\mu _o}i\)

Therefore, for Ampere’s law to be valid, the concept of displacement current is introduced.

Derivation of Expression for Displacement Current

If we zoom into the cross-section of the capacitor, there is also another kind of electric current known as displacement current due to changing electric field. Displacement current is different from the conduction current as the displacement current does not consider electron movement. The displacement current is used to study the propagation of electromagnetic waves. Displacement current is produced due to the changing electric field between the capacitor plates where Ampere’s law alone is not applicable.

Let, 

\(\Delta V\) = instantaneous potential difference across the capacitor plates

\(q\) = instantaneous charge developed on plates of a capacitor

\(C\)= capacitance of the capacitor

\({\varepsilon _o}\) = absolute permittivity

\({\varepsilon _r}\) = relative permittivity of medium between capacitor plates

\(A\)= area of capacitor plates

\(d\)= distance between capacitor plates

\(E\)= electric field developed between plates of a capacitor

\({I_D}\) = displacement current through the capacitor due to change in an electric field

The instantaneous charge developed on the capacitor plates is given by \(q = C\Delta V\)

As capacitor charges, the electric field between the capacitor plates changes. We know that: \(C = \frac{{{\varepsilon _e}{\varepsilon _i}A}}{d}\) and \(\Delta V = Ed\)

So, \(q = C\Delta V = \frac{{{\varepsilon _e}{\varepsilon _i}A}}{d}\left( {Ed} \right) = {\varepsilon _e}{\varepsilon _i}AE = {\varepsilon _o}{\varepsilon _r}{\phi _E}\)

Where \({\phi _E} = \) electric flux

We know that \({I_D} = \frac{{dq}}{{dt}} = {\varepsilon _o}{\varepsilon _r}\frac{{d\left( {{\phi _E}} \right)}}{{dt}}\)

\(\therefore {I_D} = {\varepsilon _o}{\varepsilon _r}\frac{{d\left( {{\phi _E}} \right)}}{{dt}}\)

Ampere- Maxwell’s Circuital Law

James Clerk Maxwell assumed that there must be some current between the capacitor plates during the charging process. He named it displacement current. Hence modified Ampere’s law is as follows:

\(\oint {\mathop E\limits^ \to } \mathop {dl}\limits^ \to = {\mu _o}\left( {{i_c} + {i_d}} \right)\) or \(\oint {\mathop B\limits^ \to } \mathop {dl}\limits^ \to = {\mu _o}\left( {{i_e} + {\varepsilon _o}\frac{{\varphi \varepsilon }}{{dt}}} \right)\)

Where,

\({i_c} = \) Conduction current = current due to flow of charges in a conductor

\({i_d} =\) Displacement current \( = {\varepsilon _o}\frac{{d\phi \varepsilon }}{{dt}}\)= current due to the changing electric field between the plates of the capacitor

Note:

1. Displacement current magnitude \({i_d}\) = conduction current magnitude \({i_c}\).
2. \({i_c}\) and \({i_d}\) in a circuit may not be continuous, but their sum is always continuous.

What is the Displacement Current?

The type of current produced due to changes in the electric field is called the displacement current. It varies with time according to Maxwell’s equation The S.I. unit of displacement current is Ampere (Amp). 

Maxwell’s Equations

 i. \(\oint {\mathop E\limits^ \to } \mathop {ds}\limits^ \to = \frac{q}{{{\varepsilon _o}}}\) Gauss’s law in Electrostatics  

We have learnt in Electrostatics that the total flux of electric field out of a closed surface is just the total enclosed charge multiplied by. This formula is called Maxwell’s first equation.  

ii. \(\oint {\mathop B\limits^ \to } \mathop {ds}\limits^ \to = 0\) Gauss’s law in Magnetism

The second Maxwell equation is analogous to Gauss’s law in Electrostatics. For the magnetic field, the field lines flow around in closed curves. Considering the force lines representing a kind of fluid flow, the so-called “magnetic flux”, we see that magnetic flux flows into the surface as much as it flows out for a closed surface. Therefore, the net magnetic flux out of the enclosed volume is zero. This formula is called Maxwell’s second equation.

iii. \(\oint {\mathop E\limits^ \to } \mathop {dl}\limits^ \to = \frac{{d{\phi _B}}}{{dt}}\) Faraday’s law of EMI   

In accordance with Faraday’s Second Law of Electromagnetic Induction, we have learnt that a closed circuit having a changing magnetic flux induces a circulating current, which means that there is a nonzero voltage around the circuit. This formula is called Maxwell’s third equation.

iv. \(\oint {\mathop B\limits^ \to } \mathop {dl}\limits^ \to = {\mu _o}\left( {{i_e} + {\varepsilon _o}\frac{{d\varphi \varepsilon }}{{dt}}} \right)\) Maxwell- Ampere’s Circuital law

The fourth Maxwell equation is analogous to the electrostatic version of the third equation given above, but for the magnetic field, and is the modified form of Ampere’s law. This formula is called Maxwell’s fourth equation.

Summary

1. Kirchhoff’s first law is valid at each capacitor plate provided that we take the sum of conduction and displacement current.
2. The displacement current across the plates of a capacitor is the same as the conduction current.
3. Displacement current is a current which is produced in the region of change in the electric field.
4. Maxwell- Ampere’s Circuital law, is given by \(\oint {\mathop B\limits^ \to } \mathop {dl}\limits^ \to = {\mu _o}\left( {{i_e} + {\varepsilon _o}\frac{{d\varphi \varepsilon }}{{dt}}} \right)\)

FAQs on Displacement current and Maxwell’s Equations

Q.1. What is the S.I. unit of displacement current?
Ans: The SI unit of displacement current is Ampere (A).

Q.2. What is displacement current?
Ans: Displacement current is the current due to the changing electric field.

Q.3. How does the displacement current arise?
Ans: Displacement current arises due to varying e.m.f and hence varying electric field.

Q.4. Does the displacement current satisfy the property of continuity?
Ans: Yes, the sum of conduction current and displacement current remains constant along the closed path.

Q.5. Write the expression for the displacement current.
Ans: The expression for the displacement current is \(I_D=\varepsilon_o\varepsilon_r\frac{\operatorname d(\phi_E)}{\operatorname dt}\).

Q.6. What is the displacement current when the electric fields are steady in a conducting wire?
Ans: The magnitude of displacement current in the case of steady electric fields in a conducting wire is zero.

Q.7. State Ampere-Maxwell law.
Ans: The integral of the magnetic field around a closed loop is equal to \({\mu _o}\) times the sum of conduction current and displacement current.

Q.8. Distinguish between conduction current and displacement current.
Ans: Conduction current is due to the flow of electrons in an electric circuit. It exists even if electrons flow at a steady rate. It obeys Ohm’s law. Displacement current arises due to the time-varying electric field between capacitor plates. It does not exist when a steady current flows in a region. It doesn’t obey Ohm’s law.