What are time-varying fields?

In electromagnetics, due to static electric charges, the electrostatic fields are usually produced. And due to the motion of electric charges with uniform velocity, the magnetostatic fields are produced. The field created due to accelerated charges or the time-varying current is called time-varying fields. The field produced by the time-varying currents is called electromagnetic fields. According to Faraday's law of electromagnetic fields, in any closed circuit, the induced emf is proportional to the rate of change of flux within the circuit. The below equation shows Faraday's law,

$V_{emf} = \frac{-de}{dt} = -N\frac{d\psi }{dt}$ 

where, N = number of turns in the circuit and $\psi$ = flux through each turn.

The negative sign shows that the voltage induced acts always opposite to the flux which produces it. This is known as Lenz's law. According to this law, in electromagnetics, the current flow direction is such that the magnetic field produced by the current induced will oppose the original magnetic field.

Transformer and Motional EMF

There is a connection between magnetic fields and electric fields,, so let us examine how Faraday's law links electric fields and magnetic fields. With a single turn N = 1, circuit the equation is:

$V_{emf}= -N\frac{d\psi }{dt}$.

In terms of the electric and magnetic field, it can be given by the equation:

$V_{emf} = {\oint }_L E.dl = -\frac{d }{dt} {\int }_{S}B.dS$

where $\psi$ is replaced by ${\int }_{S}B.dS$ and S = surface area of the circuit which is bounded by closed path L. 

From the above equation, it is clear that it is a time-varying situation that is both magnetic fields and electric fields are present and are co-related.  It should be kept in mind that the 'dl' and 'dS' are written in accordance with the right-hand rule and as well as Stoke's theorem. There are three ways how the variation of flux can be caused and they are: 

1. Stationary loop in a time-varying magnetic field (Transformer EMF)

In electromagnetics, when a stationary loop is in a time-varying field then the equation of emf is given by: $V_{emf}= {\oint }_S E.dl = -{\int }_S\frac{\partial B}{\partial t}.dS$

In a stationary loop, the emf which is induced by the time-varying fields is called transformer emf. Applying Stokes theorem, the above equation becomes:

${\int }_S(\nabla \times E).dS = - {\int }_S\frac{\partial B}{\partial t}.dS$

For the integrals to be equal, the integrands must also be equal so,

$\nabla \times E = -\frac{\partial B}{\partial t}$, and this equation is one of the Maxwell equations for time-varying fields.

This shows that the time-varying electric field is not conservative and is given by $\nabla \times E \ne 0$.

The principle of energy conservation is not violated.

2. Moving loop in a static magnetic field (Motional emf)

An emf is induced in the loop whenever it moves in a static magnetic field. The force on a charge moving with velocity "u" in a magnetic field "B" is given by the equation:

$F_m= Qu X B$.

The motional electric field is given by,

$E_m=\frac{F_m}{Q}= u X B$.

Considering a loop consisting of a large number of free electrons moving with velocity "u", the emf induced in the loop is given by,

$V_{emf }= {\oint }_L E_m.dl = {\oint }_L(u X B). dl$

and this emf is called  motional emf because it is caused due to the motional action.

This emf is found in motors, generators, and alternators.

3. Moving loop in time-varying magnetic field

Consider a loop moving in a time-varying magnetic field. Here, both types of EMFs, motional and transformer, are present. Combining both these EMFs, we get an equation:

$V_{emf }= {\oint }_{L}E.dl = -{\int }_S\frac{\partial B}{\partial t}.dS + {\oint }_L(u X B).dl$.

So,

 $$\begin{gathered} \nabla \times E_m= \nabla \times (u X B) \\ \Rightarrow \nabla \times E = -\frac{\partial B}{\partial t}+\nabla \times (u X B) \end{gathered}$$

Displacement current

In electromagnetics, it is known that for static electric fields, $\nabla \times H = J$ but the divergence of the curl of any vector field is equal to zero.

Consequently, $\nabla ·(\nabla \times H) = 0 = \nabla ·J$.

The continuity of currents requires that $\nabla ·J\ne 0$.

These two equations are incompatible for time-varying conditions so the static electromagnetic field equation must be modified, $\nabla \times H = J + J_d$

where $J_d$ needs to be defined.

Again, it is known that the divergence of the curl of any vector is zero so we can write it as, $\nabla ·(\nabla \times H) = 0 = \nabla ·J + \nabla ·J_d$.

For this equation to agree with the continuity current equation written above,

 $$\begin{gathered} \nabla ·J_d= -\nabla ·J = \frac{\partial v}{\partial t}= \frac{\partial }{\partial t}(\nabla ·D) = \nabla ·\frac{\partial D}{\partial t} \\ or, J_d= \frac{\partial D}{\partial t} \end{gathered}$$

Substituting this in the current continuity equation, $\nabla \times H = J + \frac{\partial D}{\partial t}$

For a time-varying field, this is the Maxwell equation.

Here, the term $J_d= \frac{\partial D}{\partial t}$ is called the displacement current density, and J is known as the conduction current density. One of the major contributions of the Maxwell field is the insertion of the term $J_d$. The electromagnetic wave propagation would not be possible without this term. Displacement current is defined based on the displacement current density which is given by,

$I_d= \int J_d . dS = \int \frac{\partial D}{\partial t} . dS$

Equation of continuity for time-varying fields

The point form of equation of continuity is given by the equation,

$\nabla ·J = -{\rho }_v$ where J = conduction current density, ${\rho }_v$ = volume charge density and $\nabla$ = vector differential operator and is given by :

$\nabla = a_x\frac{\partial }{\partial a}+a_y\frac{\partial }{\partial y}+a_z\frac{\partial }{\partial z}$.

Maxwell's equations for static electromagnetic field

Maxwell's equations for time-varying fields

Maxwell's equation in differential form are given as :

$$\begin{gathered} \nabla \times H = \frac{\partial D}{\partial t}+ J \\ \nabla \times E = -\frac{\partial B}{\partial t} \\ \nabla ·D = {\rho }_v \\ \nabla ·B = 0 \end{gathered}$$

Here, H = magnetic field strength, D = electric flux density, J = conduction current density, E = Electric field, B = magnetic flux density.

Maxwell's equation in integral form is given as,

$$\begin{gathered} {\oint }_{L}H.dL = {\int }_S(D+J).dS \\ {\oint }_{S}E.dL = -{\oint }_{S}B.dS \\ {\oint }_{S}D.dS = {\oint }_S{\rho }_v.dV \\ {\oint }_S B.dS = 0 \end{gathered}$$

Context and Applications

This topic is significant in the professional  exam for Undergraduate, Graduate, and Postgraduate courses:

• Bachelors of Technology in Electrical Engineering
• Masters of Technology in Electrical Engineering

Practice Problems

Question 1: How are electrostatic fields produced?

1. Due to static electric charges.
2. Due to motion of electric charges.
3. Both a and b.
4. None of them.

Answer: Option a

Explanation: The electrostatic fields are produced only due to static electrical charges.

Question 2: How are magnetostatic fields produced?

1. Due to static electric charges.
2. Due to motion of electric charges.
3. Due to magnetic field.
4. None of the above.

Answer: Option b

Explanation: The magnetostatic fields are produced only due to the motion of electrical charges.

Question 3: How time-varying fields are produced?

1. Motion of charges
2. Static electric charges
3. Accelerated charges
4. None of the above

Answer: Option c

Explanation: The field created due to accelerated charges or the time-varying current is called time-varying fields.

Question 4: What is induced in a coil whenever it moves in a static magnetic field?

1. EMF
2. Charge
3. Current
4. None of the above

Answer: Option a

Explanation: An emf is induced in the loop whenever it moves in a static magnetic field by the process of induction.

Question 5: What is the value of divergence of the curl of any vector field?

1. zero
2. 1
3. 2
4. Infinite

Answer: Option a

Explanation: The divergence of the curl of any vector field is equal to zero that is the curl of a gradient is a zero vector.

Related Concepts

• Electromagnetism
• Electromagnetic radiation
• Maxwell's equations