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Magnetic Field and Forces

Source lecture(s): SC134 Lec9-10

Intuition

Moving charges feel magnetic forces; currents create magnetic fields that curl around them.

Formal Definition

Lorentz force: \(\vec{F} = q\vec{v}\times\vec{B}\) Force on current element: \(d\vec{F} = i\,d\vec{l}\times\vec{B}\)

Mathematical Formulation

Biot-Savart: $\(d\vec{B} = \frac{\mu_0}{4\pi}\frac{i\,d\vec{l}\times\hat{r}}{r^2}\)$ Ampere: $\(\oint \vec{B}\cdot d\vec{l} = \mu_0 i_{\text{enc}}\)$

Derivation

Biot-Savart integrates current-element contributions; Ampere's law follows from symmetry of \(\vec{B}\) around steady currents.

Worked Example

Straight infinite wire: \(B = \mu_0 i/(2\pi r)\).

Common Mistakes

  • Using Coulomb-like \(1/r^2\) without cross product for Biot-Savart.
  • Forgetting \(\mu_0\) vs \(1/(4\pi\varepsilon_0)\).

Quiz

Q1. What direction is force on a positive charge moving right in a field into the page?

Answer

Upward (\(+\hat{j}\)).