Comparison Between Electric and Magnetic Forces

Electric forces and magnetic forces are the two fundamental forces experienced by charged particles. Although both arise from electromagnetic interactions, they differ sharply in origin, direction, and magnitude. This note derives the ratio of electric force to magnetic force between two moving charges and shows why electric force dominates in most everyday situations.

Coulomb's Law (Electric Force)

Biot–Savart Law → Magnetic Force Between Moving Charges

Using I dl = qv, the magnetic force between two moving charges is derived as:

Step 1: Write Coulomb's law for the electric force between two charges.

Step 2: Write the magnetic force between two moving charges using the Biot–Savart law with I dl = qv.

Step 3: Divide F_e​ by F_m​: \[\frac{F_e}{F_m}=\frac{1/\varepsilon_0}{\mu_0}\cdot\frac{1}{v_1v_2}\]

Step 4: Use the relation c² = \[\frac{1}{\mu_0\varepsilon_0}\]​ to simplify: \[{\frac{F_e}{F_m}=\frac{c^2}{v_1v_2}}\]

Given: Drift velocity of electrons in a conductor, v₁ = v₂ ≈ 10⁻⁵ m/s.
Find: Ratio F_e / F_m.

Solution: \[\frac{F_e}{F_m}=\frac{(3\times10^8)^2}{(10^{-5})^2}\approx10^{27}\]

Conclusion: Electric force is about 10²⁷ times stronger than magnetic force at drift-velocity scales, confirming that magnetic effects are only noticeable when large numbers of charges move collectively (as in a current-carrying wire).

Think of electric force as a shout in a quiet room — always loud and dominant — while magnetic force is a whisper that only becomes audible when thousands of people whisper together (i.e., when large currents flow, even though each individual electron's magnetic contribution is negligible).