The similar 1/𝑟3 scaling of the electric dipole field and the gravitational tidal field is not a coincidence. In both cases it arises from taking the difference of two 1/𝑟2 fields with a slight displacement ≪ 𝑟, either between two opposing charges in the source (electric dipole) or between two measurement points in the distant field (gravitational tide).
To restate: In electromagnetism, one rarely has large unbalanced charges; the field at large distances is usually created by two (or more) opposing charges slightly separated from one another. But the field at that distant location can be directly felt by a small "test charge". By contrast, with gravity you typically do have large isolated masses producing fields at large distances. However, you cannot directly measure this field with an individual "test mass", since that mass will accelerate along with the rest of the measuring apparatus. Instead, you must measure the relative force between two (or more) test masses slightly displaced from one another. Because the underlying force equations for electromagnetism and gravity treat "source" and "measuring" charges or masses symmetrically, the math is the same for both cases.
Quantitatively, suppose you have a pair of charges ±𝑄 separated by a displacement 𝒔 (with the positive charge at the "head" of the displacement vector), and you measure the field using a charge 𝑞 at a position 𝒓, where 𝑟 ≫ 𝑠. The force you measure will be:
Now, suppose you have a mass 𝑀, and you measure the tidal field with a pair of small masses 𝑚 with a relative displacement 𝒅 at a position 𝒓, where 𝑟 ≫ 𝑑. Then the relative force you measure (on the mass at the "head" of the displacement vector) will be:
Note that the magnitude of the force goes as 𝐺𝑀𝑚𝑑/𝑟3, multiplied by a dimensionless geometric factor. Dividing by 𝑑 to get the tidal force (gradient), and by 𝑚 to get the tidal field, we have our earlier result that 𝑔′ ∼ 𝐺𝑀/𝑟3.
| ← Back | Start | Next → |