Two dielectric slabs of dielectric constant K and 2K are placed inside a parallel-plate capacitor as shown. The separation between the plates is d. The thickness of both slabs is $\frac{\text{d}}{\text{2}}.$ The slab of dielectric constant 2K covers half the area of the left plate, while the slab of dielectric constant K covers the entire area of the right plate. The remaining space is vacuum. The capacitor is now charged such that the potential difference between the plates is $\Delta \text{V.}$ The magnitude electric field at the points M, N, O and P is ${\text{E}}_{\text{M}}{\text{,E}}_{\text{N}}{\text{,E}}_{\text{O}}{\text{\hspace{0.17em}and\hspace{0.17em}E}}_{\text{P}}\text{.}$ Which of the following statements is/are correct?

 

When dielectric slabs that cover only part of the area of the capacitor plates are introduced, the charge density on the capacitor plates is no longer uniform. If the charge density on the lower half of the plates (i.e. in front of the points O and P) is ${\sigma }_1$ and $-{\sigma }_1,$ then $E_0=\frac{{\sigma }_1}{2{\epsilon }_0}$ and $E_P=\frac{{\sigma }_1}{2\left(K{\epsilon }_0\right)}$

If the charge density on the upper half of the plates (i.e. in front of the points M and N) is ${\sigma }_{21}$ and $-{\sigma }_2$ then $E_M=\frac{{\sigma }_2}{2\left(2K{\epsilon }_0\right)}$and $E_N=\frac{{\sigma }_2}{2\left(K{\epsilon }_0\right)}$

Now, the potential difference between the plates can be evaluated in two ways: $\Delta V=E_0\left(\frac{d}{2}\right)+E_P\left(\frac{d}{2}\right)=\frac{{\sigma }_{1}d}{4{\epsilon }_0}(1+\frac{1}{K})$ and $\Delta V=E_M\left(\frac{d}{2}\right)+E_N\left(\frac{d}{2}\right)=\frac{3{\sigma }_{1}d}{8K{\epsilon }_0}$

Equating these expressions, we get ${\sigma }_1=\left(\frac{3}{2(K+1)}\right){\sigma }_2$

Therefore, $\frac{E_P}{E_0}=\frac{1}{K};\frac{E_M}{E_0}=\frac{{\sigma }_2}{2K{\sigma }_1}=\left(\frac{K+1}{3K}\right);\frac{E_N}{E_M}=2$

$\frac{\Delta V}{E_{0}d}=\frac{K+1}{2K}$