Two parallel infinite line charges with linear charge densities `+lambda C//m` and `-lambda` C/m are placed at a distance of 2R in free space. What is the electric field mid-way between the two line charges?

Options

A. zero
B. `(2 lambda)/(pi epsi_(0)R)`
C. `(lambda)/(pi epsi_(0)R)`
D. `(1)/(2 pi epsi_(0)R)`

Answer

To find the electric field midway between two parallel infinite line charges with linear charge densities \( +\lambda \) C/m and \( -\lambda \) C/m, we can follow these steps:

### Step 1: Understand the Configuration
We have two infinite line charges:
- The first line charge has a positive linear charge density \( +\lambda \) C/m.
- The second line charge has a negative linear charge density \( -\lambda \) C/m.
- The distance between the two line charges is \( 2R \).

### Step 2: Identify the Midpoint
The midpoint between the two line charges is at a distance of \( R \) 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:
\[ E = \frac{2k\lambda}{r} \]
where \( k \) is Coulomb's constant, \( k = \frac{1}{4\pi\epsilon_0} \).

#### Electric Field from the Positive Line Charge
At the midpoint (distance \( R \) from the positive line charge):
\[ E_{+} = \frac{2k\lambda}{R} \]
This electric field is directed away from the positive line charge (repulsive).

#### Electric Field from the Negative Line Charge
At the midpoint (distance \( R \) from the negative line charge):
\[ E_{-} = \frac{2k\lambda}{R} \]
This electric field is directed towards the negative line charge (attractive).

### Step 4: Determine the Direction of the Electric Fields
Since both electric fields are directed towards the same direction (let's assume the positive line charge is on the left and the negative line charge is on the right), they will add up.

### Step 5: Calculate the Net Electric Field
The net electric field \( E_{net} \) at the midpoint is the sum of the magnitudes of the two electric fields:
\[ E_{net} = E_{+} + E_{-} = \frac{2k\lambda}{R} + \frac{2k\lambda}{R} = \frac{4k\lambda}{R} \]

### Step 6: Substitute the Value of \( k \)
Substituting \( k = \frac{1}{4\pi\epsilon_0} \):
\[ E_{net} = \frac{4 \cdot \frac{1}{4\pi\epsilon_0} \lambda}{R} = \frac{\lambda}{\pi\epsilon_0 R} \]

### Step 7: Final Result
The electric field midway between the two line charges is:
\[ E_{net} = \frac{\lambda}{\pi\epsilon_0 R} \hat{i} \]