1. Field due to two identical infinite charged sheets

For one infinite sheet with surface charge density $\sigma$, the electric field magnitude on either side is $E=\frac{\sigma}{2\varepsilon_0}$ and it is perpendicular to the sheet.

For two identical infinite sheets carrying equal charge densities, the net field is found by superposition:

• Outside the pair, the fields due to both sheets are in the same direction, so they add.
• Between the sheets, the fields are in opposite directions, so they cancel.

Thus, $E_{\text{outside}}=\frac{\sigma}{\varepsilon_0}, \qquad E_{\text{between}}=0.$

From the figure, point $A$ is in the outside region and point $B$ is in the region between the sheets. Therefore, $|\vec E_A| > |\vec E_B|.$

2. Field due to a uniformly charged spherical body

A spherical body with uniform volume charge density $\rho$ has electric field:

• Inside the sphere at distance $r$ from center: $E_{\text{in}}=\frac{\rho r}{3\varepsilon_0}$ So the field depends on $r$.

• Outside the sphere at distance $r$ from center: $E_{\text{out}}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$

From the figure, points $C$ and $D$ are at different locations relative to the sphere such that their distances from the center are different, so their electric fields cannot be equal. Hence, $\vec E_C \ne \vec E_D.$

3. Check options

• A: $|\vec E_A|=|\vec E_B|$ is false; $\vec E_C>\vec E_D$ is not generally the correct stated relation from the figure.
• B: $\vec E_A=\vec E_B$ false, $\vec E_C=\vec E_D$ false.
• C: $\vec E_C\ne \vec E_D$ and $\vec E_A>\vec E_B$ — correct.
• D: $\vec E_A>\vec E_B$ true, but $\vec E_C=\vec E_D$ false.

Therefore, the correct option is $\boxed{\text{C}}$

4. Comparison with stored answer

Stored correct answer: $\text{C}$

My derived answer also is $\text{C}$, so they agree.