An electromagnetic wave is propagating in free space. Identify the correct situation out of the following
\(\oint \vec{E}\cdot \vec{dl}=V_{emf}\)
What is special about a propagating wave. In a travelling electromagnetic wave the fields are time-varying, so the electric field is no longer conservative: it is generated partly by the changing magnetic field. That is Faraday's law.
Faraday's law in both forms.
\(\nabla\times \vec{E}=-\dfrac{\partial \vec{B}}{\partial t} \qquad \Longleftrightarrow \qquad \oint \vec{E}\cdot d\vec{l}=-\dfrac{d\Phi_B}{dt}=V_{emf}\)
Because the magnetic flux threading any loop in the path of the wave changes continuously with time, the closed line integral of E around that loop is non-zero and equals the induced emf. Option 3 is therefore the correct situation.
Why options 1 and 2 fail. \(\oint \vec{E}\cdot d\vec{l}=0\) and \(\nabla\times\vec{E}=0\) are the same statement in integral and differential form: they say the field is conservative, derivable from a scalar potential alone. That is true only in electrostatics, where ∂B/∂t = 0. In a wave, B is oscillating, so the curl of E is non-zero — indeed if it were zero there would be no mechanism for E and B to regenerate each other and the wave could not propagate at all.
Why option 4 fails. The continuity equation (conservation of charge) reads
\(\nabla\cdot\vec{J}=-\dfrac{\partial \rho_v}{\partial t}\)
not \(\nabla\cdot\vec{J}=\rho_v\); the stated form is wrong even dimensionally (A/m³ on the left versus C/m³ on the right). Moreover, free space is charge-free and current-free, so \(\rho_v=0\) and \(\vec{J}=0\) there anyway.
The wider picture. In free space Maxwell's four equations reduce to \(\nabla\cdot\vec{E}=0\), \(\nabla\cdot\vec{B}=0\), \(\nabla\times\vec{E}=-\partial\vec{B}/\partial t\) and \(\nabla\times\vec{B}=\mu_0\varepsilon_0\,\partial\vec{E}/\partial t\). It is the last two — the mutual induction of E by changing B and of B by changing E — that combine into the wave equation and give the speed \(c=1/\sqrt{\mu_0\varepsilon_0}\).
Hence, the correct situation is \(\oint \vec{E}\cdot d\vec{l}=V_{emf}\), i.e. Faraday's law.
Similar Questions
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 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?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
