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
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