"Time-varying magnetic field will always produce an electric field". The given statement is true for:
Maxwell's third equation
Maxwell's Third Equation and Electromagnetic Induction
The statement "Time-varying magnetic field will always produce an electric field" describes a fundamental principle of electromagnetism. This phenomenon is known as electromagnetic induction, and it is precisely what Maxwell's third equation, also famously known as Faraday's Law of Induction, explains.
Faraday's Law of Induction: Maxwell's Third Equation
Faraday's Law of Induction states that a changing magnetic field through a closed loop will induce an electromotive force (EMF) in that loop, which in turn drives an induced electric current if the loop is part of a complete circuit. This induced EMF is directly related to the rate of change of magnetic flux through the loop. The presence of this induced EMF implies the existence of an induced electric field.
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Mathematical Form (Integral Form):
The integral form of Faraday's Law is given by:
\(\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{d}{dt} \iint_S \mathbf{B} \cdot d\mathbf{S}\)
Here:
- \(\oint_C \mathbf{E} \cdot d\mathbf{l}\) represents the electromotive force (EMF) around a closed path \(C\). It shows that a non-conservative electric field \(\mathbf{E}\) is generated.
- \(-\frac{d}{dt}\) denotes the time derivative.
- \(\iint_S \mathbf{B} \cdot d\mathbf{S}\) represents the magnetic flux \(\Phi_B\) through a surface \(S\) bounded by the path \(C\).
- \(\mathbf{B}\) is the magnetic field.
- The negative sign (Lenz's Law) indicates that the induced electric field (and current) will oppose the change in magnetic flux that produced it.
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Mathematical Form (Differential Form):
Using Stokes' theorem, the integral form can be converted into the differential form:
\(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)
This differential equation clearly shows that a time-varying magnetic field (\(\frac{\partial \mathbf{B}}{\partial t}\)) is the source of a curling or rotational electric field (\(\nabla \times \mathbf{E}\)). This precisely answers the question: a time-varying magnetic field produces an electric field.
Understanding Other Maxwell's Equations
To provide context, let's briefly look at what the other Maxwell's equations describe:
| Equation Number | Name | Description |
|---|---|---|
| Maxwell's First Equation | Gauss's Law for Electricity | Relates the electric field to its sources, which are electric charges. It states that electric field lines originate from positive charges and terminate on negative charges. |
| Maxwell's Second Equation | Gauss's Law for Magnetism | States that there are no isolated magnetic monopoles; magnetic field lines always form closed loops. The net magnetic flux through any closed surface is zero. |
| Maxwell's Third Equation | Faraday's Law of Induction | Describes how a time-varying magnetic field produces an electric field (as discussed above). |
| Maxwell's Fourth Equation | Ampere's Law with Maxwell's Correction | Relates the magnetic field to its sources: electric currents (conduction current) and time-varying electric fields (displacement current). This correction was crucial for predicting electromagnetic waves. |
Based on the analysis, the statement "Time-varying magnetic field will always produce an electric field" is a direct consequence and description of Maxwell's third equation, or Faraday's Law of Induction.
Important Questions from Maxwell's Equations
∇ × H = J is differential form of
Maxwell's divergence equation for the magnetic field is given by _______.
If flux density is represented by 'B' and magnetic field is represented by 'H' in a magnetic circuit, then what will be the energy density in the magnetic field?
Maxwell's third equation is derived from _______.
Which law is represented by the given expression?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
