Read the following passage and answer the questions that follow : Antennas have become increasingly importance to the society and at present, they are indispensable. They are being used every places. They are available in vast varieties. They are operating at various frequencies which are depending on different application. They operate on the principle of Maxwell's equation. They have different types of radiation patterns. There are several atmospheric losses in the way of propagation of waves. Due to which signal fades down, when it travels from transmitter to receiver antennas. Based on the above para, answer the following questions :
\(\nabla\times\nabla\times\vec{A}\) is equal to :
\(\nabla\left(\nabla\cdot\vec{A}\right)-\nabla\cdot\nabla\vec{A}\)
This is the curl-of-curl identity, and it is the single most used vector identity in electromagnetics:
\(\nabla\times\nabla\times\vec{A}=\nabla\left(\nabla\cdot\vec{A}\right)-\nabla^{2}\vec{A}\)
Since \(\nabla^{2}\vec{A}=\nabla\cdot\nabla\vec{A}\), that is option 4.
A dimensional check on the operators disposes of the others. The left-hand side is a vector, being the curl of a vector, so the right-hand side must be a vector too.
| Option | First term | Verdict |
|---|---|---|
| 1 | \(\nabla\cdot\vec{A}\) is a scalar | ✗ Scalar minus vector is meaningless |
| 2 | \(\nabla\cdot\left(\nabla\times\vec{A}\right)\) is a scalar — and identically zero | ✗ |
| 3 | Restates the left side with a constant | ✗ Not an identity at all |
| 4 | \(\nabla\left(\nabla\cdot\vec{A}\right)\) is a vector | ✓ |
Note the distinction that options 1 and 4 turn on: \(\nabla\cdot\vec{A}\) is the divergence, a scalar, but \(\nabla\left(\nabla\cdot\vec{A}\right)\) is the gradient of that scalar, which is a vector. The extra \(\nabla\) is what makes option 4 legitimate.
Why this identity matters, and why it appears under a passage about antennas. It is the step that turns Maxwell's equations into the wave equation. Taking the curl of Faraday's law and substituting Ampère's law gives
\(\nabla\times\nabla\times\vec{E}=-\mu\varepsilon\dfrac{\partial^{2}\vec{E}}{\partial t^{2}}\)
and applying the identity, with \(\nabla\cdot\vec{E}=0\) in a source-free region, collapses the left side to \(-\nabla^{2}\vec{E}\), leaving
\(\nabla^{2}\vec{E}=\mu\varepsilon\dfrac{\partial^{2}\vec{E}}{\partial t^{2}}\)
— the wave equation, from which the velocity \(1/\sqrt{\mu\varepsilon}\) follows. Without this identity there is no wave, and no radiation for an antenna to launch.
Hence, ∇×∇×A = ∇(∇·A) − ∇·∇A.
Similar Questions
A scalar function V is given by V = 2xyz2. The gradient of V is given by:
Assertion (A) : Curl of any vector is a vector. It gives normal vector which is perpendicular to both the plane and the parent vector.
Reason (R) : The value of curl of H can be found by the expression :
(Curl H)Normal = \(\lim_{\Delta S\to0}\dfrac{\oint \vec{H}\cdot d\vec{l}}{\Delta S}\)
where $\Delta S$ is the planar area and $\overline{\text{dl}}$ is the line element.
Select your answer using the codes given below.
Match the following :
List - I List - II (a) Curl operator (i) Gradient (b) Del operator (ii) Volume to surface conversion (c) Divergence tdeorem (iii) Surface to line conversion (d) Stokes tdeorem (iv) Rotation Codes :
In cylindrical coordinates, the Laplace equation holds the following expression :
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