Antennas are used for receiving and transmitting the electromagnetic signals. Their size depends upon the operating frequency / wavelength. Higher is the frequency, lower is the size of antenna. They work on Maxwell equations for field theory. They are of various types for different applications like TV transmission, AM transmission, FM transmission and satellite transmission. The waves travel in free space.
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 :
Vector Helmholtz equation
Recognise the form. Rearranged, the given equation reads
\(\nabla^{2}\mathbf{E}_S+k^{2}\mathbf{E}_S=0\)
This is the Helmholtz equation, and because the unknown is a vector field it is the vector Helmholtz equation. Option 1.
Where it comes from. Take the curl of Faraday's law and substitute Ampère's law in a source-free, lossless, linear medium:
\(\nabla\times\nabla\times\mathbf{E}=-j\omega\mu(\nabla\times\mathbf{H})=-j\omega\mu(j\omega\epsilon\mathbf{E})=\omega^{2}\mu\epsilon\mathbf{E}\)
Using the identity \(\nabla\times\nabla\times\mathbf{E}=\nabla(\nabla\cdot\mathbf{E})-\nabla^{2}\mathbf{E}\) with \(\nabla\cdot\mathbf{E}=0\) in a charge-free region gives exactly the equation above, with \(k^{2}=\omega^{2}\mu\epsilon\).
Why the phasor subscript matters. The time-harmonic assumption \(\mathbf{E}(t)=\mathrm{Re}\{\mathbf{E}_Se^{j\omega t}\}\) replaces every \(\partial/\partial t\) by jω, which is what turns the full wave equation — second order in both space and time — into an equation in space alone. Helmholtz is the frequency-domain form of the wave equation.
Distinguish it from the three distractors.
| Equation | Form | Describes |
|---|---|---|
| Vector Helmholtz | \(\nabla^{2}\mathbf{E}+k^{2}\mathbf{E}=0\) | time-harmonic wave propagation |
| Poisson | \(\nabla^{2}V=-\rho/\epsilon\) | electrostatic potential with charge; scalar, no k, has a source term |
| Laplace | \(\nabla^{2}V=0\) | Helmholtz with k = 0, the static limit |
| Diffusion | \(\nabla^{2}\mathbf{E}=j\omega\mu\sigma\mathbf{E}\) | fields in a good conductor — first order in time, the skin-effect equation |
The structural clue. Poisson's equation has a source term on the right and no k, so it cannot be a wave equation. The diffusion equation carries a factor of j on the right-hand side, giving exponential decay rather than propagation. The Coulomb gauge is a condition on the vector potential, \(\nabla\cdot\mathbf{A}=0\), not a field equation at all. Only Helmholtz has the pure \(+k^{2}\) term that produces travelling-wave solutions \(e^{-jkz}\).
Hence, the equation is the vector Helmholtz equation.
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?
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?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
