Coulomb’s Law & Electric Fields

Coulomb’s Law

• Governs the magnitude of the electrostatic force $F_e$ between two point charges.
• Scalar form (magnitude only):
  • $F<em>e = k\frac{q</em>1 q_2}{r^2}$
• Variable definitions:
  • $F_e$ – magnitude of electrostatic force (newtons, $\text{N}$)
  • $k$ – Coulomb’s (electrostatic) constant
  • In SI: $k = \frac{1}{4\pi \epsilon_0}$
  • Numeric value: k \approx 8.99\times10^9\,\text{N·m}^2\text{/C}^2
  • $q<em>1,\;q</em>2$ – magnitudes of the two interacting charges (coulombs, $\text{C}$)
  • $r$ – center-to-center distance between the charges (metres, $\text{m}$)
• Permittivity of free space (vacuum):
  • \epsilon_0 \approx 8.85\times10^{-12}\,\text{C}^2\text{/N·m}^2
• Direction rules (vector form):
  • Like charges (both $+$ or both $-$): forces are repulsive.
  • Unlike charges ($+$ with $-$): forces are attractive.
  • Force vector lies along the line joining the two charges.

Example 1 – Doubling the Distance

• Situation: A positive charge is attracted to a negative charge. Distance between them is doubled.
• Analysis with $F_e \propto \dfrac{1}{r^2}$:
  • $r \to 2r \implies r^2 \to (2r)^2 = 4r^2$
  • Therefore $F<em>{\text{new}} = \dfrac{F</em>{\text{old}}}{4}$.
• Result: Force becomes one-quarter of its original magnitude. (Exact distances/units unnecessary because ratio argument suffices.)

Analogy to Newton’s Law of Gravitation

• Coulomb: $F<em>e \propto \dfrac{q</em>1 q_2}{r^2}$
• Gravity: $F<em>g \propto \dfrac{m</em>1 m_2}{r^2}$
• Key similarity: Inverse-square dependence on distance plus proportionality to intrinsic property (charge vs. mass).
• Mnemonic: remembering one form helps recall the other on test day.

Example 2 – Electrostatic vs. Gravitational Force (Electron–Proton)

• Given data:
  • $m_p = 1.67\times10^{-27}\,\text{kg}$
  • $m_e = 9.11\times10^{-31}\,\text{kg}$
  • Elementary charge: $e = 1.60\times10^{-19}\,\text{C}$ (both electron and proton magnitude)
  • k = 8.99\times10^9\,\text{N·m}^2\text{/C}^2
  • Gravitational constant: G = 6.67\times10^{-11}\,\text{N·m}^2\text{/kg}^2
• Ratio derivation (distance $r$ cancels):
  • $\dfrac{F<em>e}{F</em>g} = \dfrac{k q<em>1 q</em>2}{G m<em>1 m</em>2}$
  • Substitute numbers:
  • Numerator: $8.99\times10^9 \times (1.60\times10^{-19})^2$
  • Denominator: $6.67\times10^{-11} \times 1.67\times10^{-27} \times 9.11\times10^{-31}$
  • Approximate math (shown in transcript): $\approx 2.27\times10^{39}$
• Interpretation: Electrostatic attraction is about $10^{40}$ times stronger than gravitational attraction for an electron–proton pair.
• Tip: Writing the symbolic ratio first causes $r^2$ to cancel, simplifying arithmetic.

Electric Field (\textit{E-field}) Basics

• Every electric charge establishes an electric field around it, analogous to a mass establishing a gravitational field.
• Purpose: describes how a charge would exert force on other charges in space without those charges necessarily being present.
• Magnitude definitions (two equivalent forms):
  1. Test-charge definition: $E = \dfrac{F<em>e}{q</em>{\text{test}}}$
  2. Source-charge definition: $E = k\dfrac{Q_{\text{source}}}{r^2}$
• Units: $\text{N/C}$ (equivalently, $\text{V/m}$ in electromagnetism).
• Derivation: Divide both sides of Coulomb’s law by $q_{\text{test}}$.

Method 1: Using a Test Charge

• Place a small “test” charge $q_{\text{test}}$ at the point of interest.
• Measure the force $F_e$ acting on it.
• Compute $E = F<em>e/q</em>{\text{test}}$.
• Limitation: Requires an actual test charge; may disturb the field or simply be absent.

Method 2: Using Source Charge Only

• No test charge required.
• Need only:
  • Magnitude of the stationary “source” charge $Q_{\text{source}}$ creating the field.
  • Distance $r$ from the source charge to the point of interest.
• Formula: $E = k Q_{\text{source}}/r^2$.

Direction of Electric Field Vectors

• Defined as the direction a positive test charge would accelerate.
• Consequences:
  • Positive source charge ($+Q$): field vectors point radially outward (repulsive to $+q$ test charge).
  • Negative source charge ($-Q$): field vectors point radially inward (attractive to $+q$ test charge).

Field-Line Representation

• Imaginary lines tangent to local $\vec{E}$ direction everywhere.
• Characteristics:
  • Originate on positive charges, terminate on negative charges.
  • Density of lines $\propto$ field strength:
  • Near charge: lines packed closely → strong field.
  • Far away: lines spread out → weak field.
  • On paper they resemble bicycle-wheel spokes.

Superposition of Electric Fields

• For multiple charges, the net electric field at any point equals the vector sum of individual fields: $\vec{E}<em>{\text{net}} = \sum</em>i \vec{E}_i$.

Force Direction vs. Test-Charge Sign

• If the test charge is positive:
  • $\vec{F}$ is parallel to $\vec{E}$.
• If the test charge is negative:
  • $\vec{F}$ is antiparallel (opposite direction) to $\vec{E}$.
• Essential for drawing correct force vectors in problems.

Practical & Conceptual Connections

• Ethical/Philosophical: None stated explicitly; focus remains on physical laws.
• Real-world relevance:
  • In atomic structure, electromagnetic forces overwhelmingly dominate gravitational forces (explains orbital electrons vs. negligible gravitational binding).
  • In engineering, field-line concepts aid capacitor design, electrostatic shielding, and safety around high-voltage equipment.
• Remembering inverse-square laws and sign conventions is critical for problem solving in electrostatics, gravitation, and optics (e.g., light intensity behaves similarly).