Prove that 1 Debye = 10⁻¹⁸ esu cm = 3.335 × 10⁻³⁰ C·m (coulomb meter).

To prove the relationship between the Debye unit and the Coulomb meter, we'll use the definitions and conversion factors for electric dipole moments.

**1 Debye = 10^-18 esu cm**

**1 Debye = 3.335 × 10^-30 C·m**

Here's the step-by-step proof:

1. **Definition of Debye:**

The Debye (D) is a unit of electric dipole moment. The definition of a Debye in terms of electrostatic units (esu) and centimeters (cm) is:
\[
1 \text{ D} = 10^{-18} \text{ esu} \cdot \text{cm}
\]

2. **Convert esu to Coulombs:**

The conversion factor between the electrostatic unit (esu) and the Coulomb is:
\[
1 \text{ esu} = \frac{1}{\sqrt{4 \pi \epsilon_0}} \text{ C} \approx 3.33564 \times 10^{-10} \text{ C}
\]

Therefore:
\[
1 \text{ esu} = 3.33564 \times 10^{-10} \text{ C}
\]

3. **Substitute and Convert:**

Given \( 1 \text{ D} = 10^{-18} \text{ esu} \cdot \text{cm} \), convert esu to Coulombs:
\[
1 \text{ D} = 10^{-18} \text{ esu} \cdot 3.33564 \times 10^{-10} \text{ C} \cdot \text{cm}
\]

\[
1 \text{ D} = (10^{-18} \times 3.33564 \times 10^{-10}) \text{ C} \cdot \text{cm}
\]

\[
1 \text{ D} = 3.33564 \times 10^{-28} \text{ C} \cdot \text{cm}
\]

4. **Adjust for Significant Figures:**

The conversion factor given is approximately \( 3.335 \times 10^{-30} \text{ C} \cdot \text{m} \). To match units, note that:
\[
1 \text{ cm} = 10^{-2} \text{ m}
\]

\[
3.33564 \times 10^{-28} \text{ C} \cdot \text{cm} = 3.33564 \times 10^{-28} \text{ C} \cdot 10^{-2} \text{ m}
\]

\[
3.33564 \times 10^{-28} \text{ C} \cdot \text{cm} = 3.33564 \times 10^{-30} \text{ C} \cdot \text{m}
\]

Thus:
\[
1 \text{ D} = 3.335 \times 10^{-30} \text{ C} \cdot \text{m}
\]

In conclusion, the conversion is verified as:
\[
1 \text{ D} = 3.335 \times 10^{-30} \text{ C} \cdot \text{m}
\]