Two parallel infinite line charges `+lamda and -lamda` are placed with a separation distance R in free space. The net electric field exactly mid-way between the two line charges is

Options

A. zero
B. `(2lamda)/(piepsilon_(0)R)`
C. `(lamda)/(piepsilon_(0)R)`
D. `(lamda)/(2piepsilon_(0)R)`

Explanation

To find the net electric field exactly midway between two parallel infinite line charges, one with charge density \( +\lambda \) and the other with charge density \( -\lambda \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Setup**:
- We have two infinite line charges: one with positive linear charge density \( +\lambda \) and the other with negative linear charge density \( -\lambda \).
- The distance between the two line charges is \( R \).

2. **Determine the Midpoint**:
- The midpoint between the two line charges is at a distance of \( \frac{R}{2} \) from each line charge.

3. **Calculate the Electric Field due to \( +\lambda \)**:
- The electric field \( E_{+\lambda} \) due to an infinite line charge is given by the formula:
\[ E = \frac{\lambda}{2 \pi \epsilon_0 r} \]
- At the midpoint (distance \( \frac{R}{2} \) from the line charge \( +\lambda \)):
\[ E_{+\lambda} = \frac{\lambda}{2 \pi \epsilon_0 \left(\frac{R}{2}\right)} = \frac{2\lambda}{\pi \epsilon_0 R} \]

4. **Calculate the Electric Field due to \( -\lambda \)**:
- Similarly, for the line charge \( -\lambda \), the electric field \( E_{-\lambda} \) at the midpoint (also at distance \( \frac{R}{2} \)):
\[ E_{-\lambda} = \frac{-\lambda}{2 \pi \epsilon_0 \left(\frac{R}{2}\right)} = \frac{-2\lambda}{\pi \epsilon_0 R} \]

5. **Determine the Direction of the Electric Fields**:
- The electric field due to \( +\lambda \) points away from the line charge (to the right if we assume the positive line charge is on the left).
- The electric field due to \( -\lambda \) points towards the line charge (also to the right).

6. **Calculate the Net Electric Field**:
- Since both electric fields point in the same direction (to the right), we can add their magnitudes:
\[ E_{\text{net}} = E_{+\lambda} + |E_{-\lambda}| \]
\[ E_{\text{net}} = \frac{2\lambda}{\pi \epsilon_0 R} + \frac{2\lambda}{\pi \epsilon_0 R} = \frac{4\lambda}{\pi \epsilon_0 R} \]

### Final Result:
The net electric field exactly midway between the two line charges is:
\[ E_{\text{net}} = \frac{4\lambda}{\pi \epsilon_0 R} \]