Certainly! To find a specific value for $r$ relative to $d$, we need to quantitatively determine when the electric field due to the dipole dominates over the contributions from higher-order terms, such as the quadrupole or individual point charges.

Approximation for Far-Field Condition

In general, the far-field condition for a dipole is satisfied when the observation distance $r$ is much larger than the separation distance $d$. For a more quantitative approach, consider the following:

1. Dipole Field Contribution: The electric field of a dipole falls off as $1/r^3$.
2. Higher-Order Terms: For example, quadrupole contributions fall off as $1/r^4$.

A good rule of thumb is that the distance $r$ should be at least 10 times the charge separation $d$ to ensure that the dipole field dominates:

$r \geq 10d$

Calculating a Specific Example

Let's assume the far-field condition requires that the dipole term contributes at least 90% to the total field.

The relative contributions from the dipole and quadrupole terms depend on $r/d$. Without going into the complex multipole expansion calculations, a general criterion used is:

$r \geq 10d$

Example Calculation:

If $d = 1$ nm (nanometer):

$r \geq 10 \times 1 \text{ nm} = 10 \text{ nm}$

Thus, for the dipole approximation to hold, the observation point should be at least 10 nm away when $d = 1$ nm.

Conclusion

For the dipole field to dominate and represent the behavior far from the molecule, the distance $r$ should be at least $10d$. Therefore, the observation point must be at least 10 times farther than the separation distance between the charges.

Would you like more details or have any questions?

Related Questions:

1. What happens if $r$ is only slightly larger than $d$?
2. How do quadrupole fields contribute to the electric field?
3. Can this approach be applied to other molecular models?
4. What role does the permittivity of the medium play in dipole fields?
5. How does the dipole moment direction influence the field observed at a distance?

Tip:

In electrostatics, understanding the hierarchy of terms (monopole, dipole, quadrupole) is crucial when analyzing fields at different distances.