The value of k in Coulomb's law is a fundamental constant that quantifies the strength of the electrostatic force between two point charges in a vacuum, and knowing its precise magnitude is essential for solving problems in electromagnetism, engineering, and physics education Worth keeping that in mind. Took long enough..
Understanding Coulomb's Law
Coulomb's law describes how two stationary electric charges interact. The law states that the magnitude of the electrostatic force (F) between two point charges (q₁ and q₂) is directly proportional to the product of the charges and inversely proportional to the square of the distance (r) separating them:
[ F = k \frac{|q_1 q_2|}{r^2} ]
Here, k is the Coulomb constant, sometimes denoted as kₑ. Its value determines how strongly charges attract or repel each other, and it carries the units that make the equation dimensionally consistent in the International System of Units (SI).
The Coulomb Constant (k) – Definition and Value
Numerical Value in SI Units
In a vacuum, the Coulomb constant has the exact value:
[ k = 8.987,551,7923 \times 10^{9}\ \mathrm{N\cdot m^{2}/C^{2}} ]
For most classroom and practical calculations, it is rounded to:
[ k \approx 9.0 \times 10^{9}\ \mathrm{N\cdot m^{2}/C^{2}} ]
This approximation introduces an error of less than 0.14 %, which is negligible for introductory physics problems.
Relation to Vacuum Permittivity (ε₀)
The constant k is not an independent fundamental quantity; it is derived from the vacuum permittivity (ε₀), a measure of how much electric field is permitted to exist in free space:
[ k = \frac{1}{4\pi \varepsilon_{0}} ]
The CODATA 2018 value of ε₀ is:
[ \varepsilon_{0} = 8.854,187,817 \times 10^{-12}\ \mathrm{F/m} ]
Substituting this into the expression above yields the numerical value of k shown earlier. This relationship highlights that k embodies the geometric factor (4\pi) that arises from the spherical symmetry of point‑charge fields.
Approximate Value and Practical Use
Because the exact figure contains many significant digits, textbooks often present k as:
- k ≈ 9.0 × 10⁹ N·m²/C² – useful for quick estimates.
- k = 8.99 × 10⁹ N·m²/C² – a slightly more precise version still easy to remember.
When high precision is required (e.On the flip side, g. , in metrology or semiconductor design), the full CODATA value is used.
Derivation and Historical Determination
Early Experiments (Coulomb's Torsion Balance)
Charles-Augustin de Coulomb first measured the force between charged spheres in 1785 using a torsion balance. By observing the angular deflection caused by known charges at various distances, he inferred the inverse‑square dependence and obtained an empirical constant that later became k. His apparatus could not determine ε₀ directly, but it established the proportionality that underpins the modern definition It's one of those things that adds up..
Worth pausing on this one.
Modern Measurements and CODATA
Today, k is determined indirectly through highly accurate measurements of ε₀ via techniques such as:
- Capacitance bridges – measuring the capacitance of known geometries.
- Josephson effect and quantum Hall effect – linking electrical units to fundamental constants.
- Speed of light (c) and magnetic constant (μ₀) – using the relation (c^{2}=1/(\varepsilon_{0}\mu_{0})).
The Committee on Data for Science and Technology (CODATA) aggregates these results, publishing the accepted value of k (and ε₀) every few years. The current value reflects uncertainties on the order of 1 × 10⁻¹² N·m²/C², far below the needs of most applications.
Dependence on Medium – Dielectric Constant
The value of k changes when charges are immersed in a material medium because the medium polarizes, reducing the effective field. In a linear, isotropic dielectric, Coulomb's law becomes:
[ F = \frac{k}{\varepsilon_{r}} \frac{|q_1 q_2|}{r^{2}} ]
where εᵣ (epsilon‑r) is the relative permittivity or dielectric constant of the material. The modified constant is often written as:
[ k_{\text{medium}} = \frac{k}{\varepsilon_{r}} = \frac{1}{4\pi \varepsilon_{0}\varepsilon_{r}} ]
Examples of εᵣ
| Material | Approximate εᵣ (at room temperature) | Effect on k |
|---|---|---|
| Vacuum | 1 | k unchanged |
| Air | 1.0006 | Slight reduction |
| Water | ~80 | k reduced by factor 80 |
| Glass | 5–10 | k reduced 5‑ to 10‑fold |
| Silicon | 11.7 | k reduced ≈12‑fold |
Thus, the same pair of charges experiences
