Two charges `+0.2 mu mu C and -0.2 mu mu C` are placed `10^(-6) cm` apart. Calculate electric field at an axial point at a distance of 10 cm from their middle point.

Explanation

To calculate the electric field at an axial point at a distance of 10 cm from the midpoint of two charges \( +0.2 \, \mu C \) and \( -0.2 \, \mu C \) that are placed \( 10^{-6} \, cm \) apart, we can follow these steps:

### Step 1: Understand the Configuration
We have two charges:
- Charge \( q_1 = +0.2 \, \mu C = 0.2 \times 10^{-6} \, C \)
- Charge \( q_2 = -0.2 \, \mu C = -0.2 \times 10^{-6} \, C \)

The distance between the charges \( d = 10^{-6} \, cm = 10^{-8} \, m \).

The distance from the midpoint of the charges to the point where we want to calculate the electric field is \( r = 10 \, cm = 0.1 \, m \).

### Step 2: Calculate the Dipole Moment
The dipole moment \( p \) is given by the formula:
\[ p = q \cdot d \]
where \( q \) is the magnitude of one of the charges and \( d \) is the separation between the charges.

Substituting the values:
\[ p = (0.2 \times 10^{-6} \, C) \cdot (10^{-8} \, m) = 2 \times 10^{-14} \, C \cdot m \]

### Step 3: Use the Formula for Electric Field of a Dipole
The electric field \( E \) at a point on the axial line of a dipole is given by the formula:
\[ E = \frac{2kp}{r^3} \]
where \( k \) is Coulomb's constant, \( k = 9 \times 10^9 \, N \cdot m^2/C^2 \).

### Step 4: Substitute the Values into the Formula
Now substituting the values into the electric field formula:
\[ E = \frac{2 \cdot (9 \times 10^9) \cdot (2 \times 10^{-14})}{(0.1)^3} \]

Calculating \( (0.1)^3 \):
\[ (0.1)^3 = 0.001 = 10^{-3} \]

Now substituting:
\[ E = \frac{2 \cdot (9 \times 10^9) \cdot (2 \times 10^{-14})}{10^{-3}} \]

### Step 5: Simplify the Expression
Calculating the numerator:
\[ 2 \cdot (9 \times 10^9) \cdot (2 \times 10^{-14}) = 36 \times 10^{-5} = 3.6 \times 10^{-4} \]

Now substituting back into the electric field equation:
\[ E = \frac{3.6 \times 10^{-4}}{10^{-3}} = 3.6 \times 10^{1} = 36 \, N/C \]

### Final Result
Thus, the electric field at the axial point at a distance of 10 cm from the midpoint of the dipole is:
\[ E = 36 \, N/C \]