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`

Explanation

To solve the problem, we need to find the point between the two electric dipoles where the electric field is zero. Let's denote the dipole with moment `p` as Dipole 1 and the dipole with moment `64p` as Dipole 2. The distance between the two dipoles is `25 cm`.

### Step-by-Step Solution:

1. **Define the Positions of the Dipoles:**
- Let Dipole 1 (moment `p`) be at position `0 cm`.
- Let Dipole 2 (moment `64p`) be at position `25 cm`.

2. **Define the Point of Interest:**
- Let the point where the electric field is zero be at a distance `x` from Dipole 1. Therefore, the distance from Dipole 2 will be `25 cm - x`.

3. **Electric Field Due to a Dipole:**
- The electric field `E` at a distance `r` from a dipole moment `p` is given by the formula:
\[ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3} \]
- For Dipole 1 (moment `p`), the electric field at point `x` is:
\[ E_1 = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{x^3} \]
- For Dipole 2 (moment `64p`), the electric field at point `25 cm - x` is:
\[ E_2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{2 \cdot 64p}{(25 - x)^3} \]

4. **Setting the Electric Fields Equal:**
- Since the dipoles are in opposite directions, we set the magnitudes of the electric fields equal to find the point where the net electric field is zero:
\[ E_1 = E_2 \]
\[ \frac{2p}{x^3} = \frac{128p}{(25 - x)^3} \]

5. **Canceling Common Terms:**
- We can cancel `2p` from both sides:
\[ \frac{1}{x^3} = \frac{64}{(25 - x)^3} \]

6. **Cross-Multiplying:**
- Cross-multiplying gives:
\[ (25 - x)^3 = 64x^3 \]

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

8. **Solving for `x`:**
- Rearranging gives:
\[ 25 = 5x \implies x = 5 \text{ cm} \]

### Conclusion:
The distance from the dipole of moment `p` where the electric field is zero is `5 cm`.