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 :
7.5 c/m2
Gauss's law in point form is the whole method.
\(\rho_{v}=\nabla\cdot\overline{D}=\dfrac{\partial D_{x}}{\partial x}+\dfrac{\partial D_{y}}{\partial y}+\dfrac{\partial D_{z}}{\partial z}\)
Take the three partial derivatives one at a time, treating the other two coordinates as constants.
\(\dfrac{\partial}{\partial x}\left[20xy^{2}(z+1)\right]=20y^{2}(z+1)\)
\(\dfrac{\partial}{\partial y}\left[20x^{2}y(z+1)\right]=20x^{2}(z+1)\)
\(\dfrac{\partial}{\partial z}\left[10x^{2}y^{2}\right]=0\)
The last one vanishes because the z-component contains no z at all — a detail easy to miss when the expression looks long.
Add them and factorise :
\(\rho_{v}=20(z+1)\left(x^{2}+y^{2}\right)\)
Substitute the point P(0.3, 0.4, 0.5) :
\(x^{2}+y^{2}=0.09+0.16=0.25\)
\(z+1=1.5\)
\(\rho_{v}=20\times1.5\times0.25=7.5\ \text{C/m}^{3}\)
which is option 2. (The units printed as C/m2 in the options are a misprint — a volume charge density is always C/m3, and the divergence of a C/m2 quantity necessarily gains an extra inverse metre.)
| Term | Derivative | Value at P |
|---|---|---|
| Dx | 20y2(z+1) | 20(0.16)(1.5) = 4.8 |
| Dy | 20x2(z+1) | 20(0.09)(1.5) = 2.7 |
| Dz | 0 | 0 |
| Total | 7.5 | |
What the result means physically. The divergence measures how much flux springs out of a vanishingly small volume around the point, so a positive value says there is net positive charge there. Had the answer been zero, the point would have been charge-free and the field merely passing through — which is exactly the case everywhere in free space, where \(\nabla\cdot\overline{D}=0\).
A useful cross-check is that this D field is derived from a potential, so its curl should vanish; and note that using D rather than E means the answer holds whatever the permittivity of the medium, since \(\overline{D}=\varepsilon\overline{E}\) has already absorbed it.
Hence, the volume charge density at P is 7.5 C/m3.
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 :
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?
\(\int B.dl = \mu_oi_c+\mu_0\epsilon_0 \frac{d \Phi_E}{dt}\)
