GATE 2026 PH – Question 47
An infinitely large thin sheet in the xy-plane carries uniform positive charge density and is moving with constant velocity v in the +x direction (see figure below). The direction of the corresponding Poynting vector is
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Correct answer: (A) +x for both z< 0 and z> 0
Explanation
A positively charged sheet in the $xy$-plane moves with velocity $v\hat x$. It has two effects:
**Electric field.** The sheet produces $\mathbf E=\dfrac{\sigma}{2\epsilon_0}\hat z$ for $z>0$ and $\mathbf E=-\dfrac{\sigma}{2\epsilon_0}\hat z$ for $z<0$ (it points away from the positive sheet).
**Magnetic field.** The moving charge is a surface current $\mathbf K=\sigma v\,\hat x$. For a current sheet, $\mathbf B=\dfrac{\mu_0}{2}\,\mathbf K\times\hat n$, where $\hat n$ points from the sheet toward the field point.
- For $z>0$ ($\hat n=\hat z$): $\hat x\times\hat z=-\hat y$, so $\mathbf B\propto-\hat y$.
- For $z<0$ ($\hat n=-\hat z$): $\mathbf B\propto+\hat y$.
**Poynting vector** $\mathbf S\propto\mathbf E\times\mathbf B$:
- $z>0$: $\hat z\times(-\hat y)=+\hat x$.
- $z<0$: $(-\hat z)\times\hat y=+\hat x$.
Both fields reverse on crossing the sheet, so the product keeps its direction: the Poynting vector points along **$+x$ for both $z<0$ and $z>0$** (option A), in the direction of the sheet's motion.