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Subject : Electromagnetic Field Theory

please mention each step involving in the solution.

a) Vector \( \vec{P} \) is given in different coordinates as below. Find the nature of field by obtaining divergence and curl in each case as given below: \[ \begin{array}{c} \vec{P}=30 a \hat{x}+2 x y^{2} \hat{y}+5 x^{2} z^{2} \hat{z} \\ \vec{P}=\frac{150}{r^{2}} \hat{r}+5 \widehat{\emptyset} \end{array} \] b) In case of a vector field \( \vec{A}=y z \vec{a}_{\mathrm{x}}+z x \vec{a}_{\mathrm{y}}+x y \vec{a}_{z, n} \) prove that it is solenoidal. c) Consider a capacitor connected to a time varying voltage source. If we select an open surface s bounded by contour c, ampere's law suggests that \( \oint_{c} \vec{H} \cdot \overrightarrow{d l}=i \). However if we consider another surface \( S \) as the open surface bounded by the same closed contour, the current passing through this surface is zero i.e., \( \oint_{c} \vec{H} \cdot \overrightarrow{d l}=0 \) Explain the contradiction of these two equations according to Maxwell. d) Prove Kirchhoff's current law from equation of continuity.

In this part, we look at two significant procedure on a vector field: difference and twist. They are essential to the field of analytics