Two electric dipoles of moment `P` and `64P` are placed in opposite direction on a line at a distance of `25 cm`. The electric field will be zero at point between the dipoles whose distance from the dipole of moment `P` is

Options

A. 5 cm
B. `(25)/(9) cm`
C. 10 cm
D. `(4)/(13) cm`

Answer

To solve the problem of finding the point between two electric dipoles where the electric field is zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two dipoles: one with dipole moment \( P \) and the other with dipole moment \( 64P \).
- The distance between the two dipoles is \( 25 \, \text{cm} \).
- Let’s denote the distance from the dipole with moment \( P \) to the point where the electric field is zero as \( x_1 \). Consequently, the distance from the dipole with moment \( 64P \) to this point will be \( 25 - x_1 \).

2. **Electric Field Due to a Dipole**:
- The electric field \( E \) due to a dipole at a distance \( r \) is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2P}{r^3} \]
- Here, \( P \) is the dipole moment and \( r \) is the distance from the dipole.

3. **Setting Up the Equation**:
- For the dipole with moment \( P \) at distance \( x_1 \):
\[ E_1 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2P}{x_1^3} \]
- For the dipole with moment \( 64P \) at distance \( 25 - x_1 \):
\[ E_2 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2 \cdot 64P}{(25 - x_1)^3} \]

4. **Equating the Electric Fields**:
- Since the electric field is zero at the point between the dipoles, we set \( E_1 = E_2 \):
\[ \frac{2P}{x_1^3} = \frac{2 \cdot 64P}{(25 - x_1)^3} \]
- We can simplify this equation by canceling \( 2P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{1}{x_1^3} = \frac{64}{(25 - x_1)^3} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ (25 - x_1)^3 = 64 x_1^3 \]

6. **Taking the Cube Root**:
- Taking the cube root of both sides:
\[ 25 - x_1 = 4 x_1 \]

7. **Solving for \( x_1 \)**:
- Rearranging the equation:
\[ 25 = 5 x_1 \implies x_1 = \frac{25}{5} = 5 \, \text{cm} \]

### Final Answer:
The distance from the dipole of moment \( P \) to the point where the electric field is zero is \( 5 \, \text{cm} \).