Question

Two infinite plane parallel sheets, separated by a distance $d$ have equal and opposite uniform charge densities $\sigma$. Electric field at a point between the sheets is

• A. $\frac{\sigma}{2\varepsilon_0}$
• B. $\frac{\sigma}{\varepsilon_0}$
• C. zero
• D. depends on the location of the point

Hint:

This parallel sheet setup forms the core design of a standard parallel-plate capacitor! The fields cancel out to zero on the outside, but double to $\frac{\sigma}{\varepsilon_0}$ on the inside to store electrical energy.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The electric field generated by an infinite thin sheet of uniform charge is uniform and independent of the distance from the sheet. By using Gauss’s Law, we can find the field produced by an individual sheet and then apply the principle of superposition to determine the combined net field between two sheets.

Step 2: Key Formula or Approach:
1. The magnitude of the electric field ($E$) due to a single infinite sheet with charge density $\sigma$ is: $$E = \frac{\sigma}{2\varepsilon_0}$$ 2. The field points directly away from a positively charged sheet ($+\sigma$) and points directly toward a negatively charged sheet ($-\sigma$).

Step 3: Detailed Explanation:
Consider a point located in the space between the two sheets: Field due to the positive sheet ($E_+$): Since it is positively charged, its electric field vectors point straight away from it, pushing towards the negative sheet. \[ E_+ = \frac{\sigma}{2\varepsilon_0} \quad (\text{directed towards the negative sheet}) \] Field due to the negative sheet ($E_-$): Since it is negatively charged, its electric field vectors pull straight towards it, in the same direction as $E_+$. \[ E_- = \frac{\sigma}{2\varepsilon_0} \quad (\text{directed towards the negative sheet}) \] Because both field vectors point in the exact same direction in the middle region, we add their magnitudes together to find the net field: \[ E_{\text{net}} = E_+ + E_- \] \[ E_{\text{net}} = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{2\sigma}{2\varepsilon_0} = \frac{\sigma}{\varepsilon_0} \] This result is independent of the distance $d$ or the exact position between the plates, creating a perfectly uniform electric field.

Step 4: Final Answer:
The electric field at any point between the sheets is $\frac{\sigma}{\varepsilon_0}$.