Consider the following statements regarding Maxwell's equations in differential form (Symbols have their usual meanings) : (a) For free space \(\nabla\times\overline{H}=(\sigma+j\omega\epsilon)\overline{E}\) Of these statements :
(b) For free space \(\nabla\cdot\overline{D}=\rho\)
(c) For steady current \(\nabla\times\overline{H}=\overline{J}\)
(d) For static electric field \(\nabla\cdot\overline{D}=\rho\)
(c) and (d) are correct
Each statement attaches an equation to a stated condition, so both parts must agree.
(a) is wrong for the condition given. The expression \(\nabla\times\overline{H}=(\sigma+j\omega\epsilon)\overline{E}\) is the correct time-harmonic form for a general conducting medium, where \(\sigma\overline{E}\) is the conduction current and \(j\omega\epsilon\overline{E}\) the displacement current. Free space has \(\sigma=0\), so the correct free-space form is simply \(\nabla\times\overline{H}=j\omega\epsilon_{0}\overline{E}\). Quoting the lossy-medium equation as "for free space" is the error.
(b) is wrong for the same kind of reason. Free space contains no charge, so \(\rho=0\) and the free-space statement is \(\nabla\cdot\overline{D}=0\).
(c) is correct. For steady (DC) currents nothing varies with time, so the displacement term \(\dfrac{\partial\overline{D}}{\partial t}\) vanishes and Ampere's law reduces to its magnetostatic form:
\(\nabla\times\overline{H}=\overline{J}\)
(d) is correct. Gauss's law for the electric field holds whether or not the field varies with time, and is certainly valid in the static case:
\(\nabla\cdot\overline{D}=\rho\)
So (c) and (d) — option 3.
| Law | General form | Static / steady form |
|---|---|---|
| Gauss (electric) | \(\nabla\cdot\overline{D}=\rho\) | Unchanged |
| Gauss (magnetic) | \(\nabla\cdot\overline{B}=0\) | Unchanged |
| Faraday | \(\nabla\times\overline{E}=-\dfrac{\partial\overline{B}}{\partial t}\) | \(\nabla\times\overline{E}=0\) |
| Ampere | \(\nabla\times\overline{H}=\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) | \(\nabla\times\overline{H}=\overline{J}\) |
The displacement current term is the historic addition and the whole reason electromagnetic waves exist. Without it, Ampere's law \(\nabla\times\overline{H}=\overline{J}\) is mathematically inconsistent with charge conservation — taking the divergence of both sides would force \(\nabla\cdot\overline{J}=0\) always, which fails for a charging capacitor. Maxwell's insertion of \(\partial\overline{D}/\partial t\) repaired that and, with Faraday's law, produced a self-sustaining wave in which a changing E field creates H and a changing H field creates E.
Hence, the correct statements are (c) and (d).
Similar Questions
An electromagnetic wave is propagating in free space. Identify the correct situation out of the following
The unit of \(\left(\sigma E + \frac{\partial D}{\partial t}\right)\) is
As per the Maxwell's equations and their applications {Let all symbols are used with their usual meaning}
A. The tangential component of electric flux density, at a conducting surface, is a non zero quantity.
B. Electric field intensity can be found as \(E = \nabla V\).
C. For a time varying field the value of \(\oint \vec{E}\cdot \vec{dl}\) will be non-zero
D. The value of current flowing in the wire will be equal to \(\oint \vec{H}\cdot \vec{dl}\)
E. The magneto static field is conservative in nature.
Choose the correct answer from the options given below :
Which one of the fundamental equation was modified by Maxwell to form the basis of electro magnetic theory?
If Es is the field intensity vector identified as a phasor by its subscript ‘S’ and ko is the wave number, equation \(\nabla^{2}\mathbf{E}_S=-k^{2}\mathbf{E}_S\) is known as :
A medium has the value of displacement flux density
\(\overline{D}=20xy^{2}(z+1)\hat{a}_{x}+20x^{2}y(z+1)\hat{a}_{y}+10x^{2}y^{2}\hat{a}_{z}\ \text{Coulomb/m}^{2}\)
The volume charge density at a point P(0.3, 0.4, 0.5) is given by :
Match the following :
List - I List - II (a) \(\nabla\cdot\overline{D}\) (i) 0 (b) \(\nabla\cdot\overline{B}\) (ii) \(\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) (c) \(\nabla\times\overline{H}\) (iii) \(-\dfrac{\partial\overline{B}}{\partial t}\) (d) \(\nabla\times\overline{E}\) (iv) \(\rho_{v}\) Codes :
Match List – I with List – II and select the correct answer using codes given below :
List – I List – II a. \(\nabla\times\overline{H}\) i. \(\rho_{v}\) b. \(\nabla\cdot\overline{D}\) ii. \(-\dfrac{\partial B}{\partial t}\) c. \(\nabla\times\overline{E}\) iii. \(\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) d. \(\nabla\cdot\left(\nabla\times\overline{B}\right)\) iv. 0 Codes :
For a steady magnetic fields, which of the following is true :
1. The tangential component of magnetic field is continuous across any boundary except the surface of perfect conductor.
2. The tangential component of magnetic flux density is continuous across any boundary.
3. The normal component of magnetic flux density is continuous across any boundary.
4. The normal component of electric field is continuous across the boundary.Which one of the following is correct ?
Which of the following are Maxwell equations ?
1. \(B=\mu H\)
2. \(E=\dfrac{D}{\epsilon}\)
3. \(E=\dfrac{J}{\sigma}\)4. \(E=\epsilon D\)
Select the correct answer :
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?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
