GATE 2026 PH – Question 12
An infinitely large non-conducting thin sheet in the xy plane ($z=0$) carries a uniform surface charge density $\sigma=17.70\times10^{-12}\ \mathrm{C\,m^{-2}}$. The electric field in the region $z<0$ is $\mathbf E_2=\hat x+2\hat y+3\hat z$. Then, the electric field $\mathbf E_1$ in the region $z>0$ will be ($\epsilon_0=8.85\times10^{-12}\ \mathrm{C^2N^{-1}m^{-2}}$)
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Correct answer: (A) $\mathbf E_1=\hat x+2\hat y+5\hat z$
Explanation
**Boundary conditions for the electric field at a charged sheet** (normal along $z$):
- The **tangential** components of $\mathbf E$ are continuous across the sheet.
- The **normal** component jumps by $\sigma/\epsilon_0$:
$$E_{1z}-E_{2z}=\frac{\sigma}{\epsilon_0}.$$
**Surface density term:**
$$\frac\sigma{\epsilon_0}=\frac{17.70\times10^{-12}}{8.85\times10^{-12}}=2 .$$
**Field above the sheet** ($z>0$), given $\mathbf E_2=\hat x+2\hat y+3\hat z$ below:
- $E_{1x}=1$ and $E_{1y}=2$ (continuous),
- $E_{1z}=3+2=5$.
$$\mathbf E_1=\hat x+2\hat y+5\hat z\quad(\text{option A}).$$