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 solve the problem of finding the net electric field exactly midway between two parallel infinite line charges with charge densities \( +\lambda \) and \( -\lambda \) separated by a distance \( R \), we can follow these steps:

### Step 1: Understand the Configuration
We have two infinite line charges:
- The first line charge has a linear charge density of \( +\lambda \).
- The second line charge has a linear charge density of \( -\lambda \).
- The distance between the two line charges is \( R \).

### Step 2: Identify the Midpoint
The midpoint between the two line charges is located at a distance of \( \frac{R}{2} \) from each line charge.

### Step 3: Calculate the Electric Field Due to Each Line Charge
The electric field \( E \) due to an infinite line charge with linear charge density \( \lambda \) at a distance \( r \) from the line charge is given by the formula:
\[ E = \frac{\lambda}{2\pi \epsilon_0 r} \]

#### Electric Field Due to the Positive Line Charge
At the midpoint (distance \( \frac{R}{2} \)):
\[ E_{+\lambda} = \frac{\lambda}{2\pi \epsilon_0 \left(\frac{R}{2}\right)} = \frac{\lambda}{2\pi \epsilon_0} \cdot \frac{2}{R} = \frac{2\lambda}{2\pi \epsilon_0 R} \]

#### Electric Field Due to the Negative Line Charge
Similarly, for the negative line charge:
\[ E_{-\lambda} = \frac{-\lambda}{2\pi \epsilon_0 \left(\frac{R}{2}\right)} = \frac{-\lambda}{2\pi \epsilon_0} \cdot \frac{2}{R} = \frac{-2\lambda}{2\pi \epsilon_0 R} \]

### Step 4: Determine the Direction of the Electric Fields
- The electric field due to the positive line charge \( E_{+\lambda} \) points away from the line charge (to the right).
- The electric field due to the negative line charge \( E_{-\lambda} \) points towards the line charge (also to the right).

### Step 5: Calculate the Net Electric Field
Since both electric fields point in the same direction (to the right), we can add their magnitudes:
\[ E_{\text{net}} = E_{+\lambda} + |E_{-\lambda}| \]
\[ E_{\text{net}} = \frac{2\lambda}{2\pi \epsilon_0 R} + \frac{2\lambda}{2\pi \epsilon_0 R} = \frac{4\lambda}{2\pi \epsilon_0 R} \]
\[ E_{\text{net}} = \frac{2\lambda}{\pi \epsilon_0 R} \]

### Final Answer
The net electric field exactly midway between the two line charges is:
\[ E_{\text{net}} = \frac{2\lambda}{\pi \epsilon_0 R} \]