To solve this problem, we need to use the concept of electric fields due to dipoles and determine where the field cancels out between the two dipoles.
Consider two dipoles \( P \) and \( 64P \) placed on the x-axis. Let the distance between them be 25 cm, and let \( d \) be the distance from the dipole with moment \( P \) to the point where the electric field is zero. Thus, the distance from the dipole with moment \( 64P \) to this point is \( 25 - d \).
The electric field due to a dipole at a point along its axial line (assuming it's far from the dipole) is given by:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{2P}{r^3} \]
where \( P \) is the dipole moment, \( r \) is the distance from the dipole, and \( \epsilon_0 \) is the permittivity of free space.
Let’s denote the dipole with moment \( P \) as Dipole 1 and the one with moment \( 64P \) as Dipole 2.
For the electric field to be zero at a point between the dipoles, the magnitudes of the fields due to each dipole must be equal:
\[ \frac{2P}{d^3} = \frac{2 \times 64P}{(25 - d)^3} \]
Cancel the common terms and simplify:
\[ \frac{1}{d^3} = \frac{64}{(25 - d)^3} \]
Taking the cube root of both sides:
\[ \frac{1}{d} = \frac{4}{25 - d} \]
Cross-multiplying:
\[ 25 - d = 4d \]
Solving for \( d \):
\[ 25 = 5d \]
\[ d = 5 \text{ cm} \]
So the distance from the dipole of moment \( P \) to the point where the electric field is zero is 5 cm.
None of the provided options directly match this result, so there might be a need to recheck or reconsider the problem’s parameters or options.
