Match the following : Codes :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) \(\rho_{v}\)
(a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
This is simply Maxwell's four equations in point form, shuffled.
(a) \(\nabla\cdot\overline{D}=\rho_{v}\) → (iv). Gauss's law for electricity: electric flux diverges from charge, so charge is a source of the D field.
(b) \(\nabla\cdot\overline{B}=0\) → (i). Gauss's law for magnetism: magnetic flux lines always close on themselves because no magnetic monopole has ever been found. This is the one Maxwell equation with a zero on the right, and it is what makes B expressible as the curl of a vector potential, \(\overline{B}=\nabla\times\overline{A}\).
(c) \(\nabla\times\overline{H}=\overline{J}+\dfrac{\partial\overline{D}}{\partial t}\) → (ii). Ampere's law with Maxwell's displacement-current correction: a magnetic field curls around both a conduction current and a changing electric field.
(d) \(\nabla\times\overline{E}=-\dfrac{\partial\overline{B}}{\partial t}\) → (iii). Faraday's law of induction, the minus sign being Lenz's law — the induced effect always opposes the change that produced it.
The order (iv), (i), (ii), (iii) is option 1.
| Equation | Name | Physical statement |
|---|---|---|
| \(\nabla\cdot\overline{D}=\rho_{v}\) | Gauss (electric) | Charges are sources of E |
| \(\nabla\cdot\overline{B}=0\) | Gauss (magnetic) | No magnetic monopoles |
| \(\nabla\times\overline{H}=\overline{J}+\dot{\overline{D}}\) | Ampere-Maxwell | Currents and changing E make H |
| \(\nabla\times\overline{E}=-\dot{\overline{B}}\) | Faraday | Changing B makes E |
The structural pattern makes the set easy to recall. The two divergence equations describe sources: electric field has them, magnetic field does not. The two curl equations describe circulation, and they are almost mirror images of each other — a changing B produces a curling E, and a changing D produces a curling H — differing only in the minus sign and in the extra conduction term J, which exists because electric charges flow but magnetic ones do not.
Why the last two together give light. Take the curl of Faraday's law and substitute Ampere's, and in a source-free region the result is the wave equation
\(\nabla^{2}\overline{E}=\mu\varepsilon\dfrac{\partial^{2}\overline{E}}{\partial t^{2}}\)
whose velocity \(1/\sqrt{\mu_{0}\varepsilon_{0}}\) came out at 3 × 108 m/s — the calculation that told Maxwell light was an electromagnetic wave.
Hence, the correct match is (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii).
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 :
Consider the following statements regarding Maxwell's equations in differential form (Symbols have their usual meanings) :
(a) For free space \(\nabla\times\overline{H}=(\sigma+j\omega\epsilon)\overline{E}\)
(b) For free space \(\nabla\cdot\overline{D}=\rho\)
(c) For steady current \(\nabla\times\overline{H}=\overline{J}\)
(d) For static electric field \(\nabla\cdot\overline{D}=\rho\)Of these statements :
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?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
