Two thin infinite parallel plates have uniform charge densities `+ sigma` and `- sigma`. The electric field in the space between them is

Options

A. `(sigma)/(epsi_(0))`
B. `(sigma)/(2 epsi_(0))`
C. `(2sigma)/(epsi_(0))`
D. zero

Explanation

To find the electric field in the space between two infinite parallel plates with uniform charge densities \( +\sigma \) and \( -\sigma \), we can follow these steps:

### Step 1: Understand the Electric Field due to a Single Infinite Plate
The electric field \( E \) due to a single infinite plane sheet of charge with surface charge density \( \sigma \) is given by the formula:
\[ E = \frac{\sigma}{2\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Determine the Direction of the Electric Fields
For the positively charged plate with charge density \( +\sigma \), the electric field \( E_1 \) points away from the plate. For the negatively charged plate with charge density \( -\sigma \), the electric field \( E_2 \) points towards the plate.

### Step 3: Calculate the Electric Fields
Using the formula from Step 1:
- The electric field due to the positively charged plate \( E_1 \) is:
\[ E_1 = \frac{\sigma}{2\epsilon_0} \]
- The electric field due to the negatively charged plate \( E_2 \) is:
\[ E_2 = \frac{-\sigma}{2\epsilon_0} \]

### Step 4: Find the Net Electric Field Between the Plates
The net electric field \( E_{\text{net}} \) in the region between the two plates is the vector sum of the individual electric fields:
\[ E_{\text{net}} = E_1 + E_2 \]
Substituting the values from Step 3:
\[ E_{\text{net}} = \frac{\sigma}{2\epsilon_0} + \left(-\frac{\sigma}{2\epsilon_0}\right) \]
This simplifies to:
\[ E_{\text{net}} = \frac{\sigma}{2\epsilon_0} + \frac{\sigma}{2\epsilon_0} = \frac{\sigma}{\epsilon_0} \]

### Step 5: Conclusion
The electric field in the space between the two plates is uniform and given by:
\[ E_{\text{net}} = \frac{\sigma}{\epsilon_0} \]
This electric field is constant and does not depend on the distance between the plates.

### Final Answer
The electric field in the space between the two plates is:
\[ E = \frac{\sigma}{\epsilon_0} \]

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