Magnetic Field from an Infinite Current-Carrying Wire

Current Through the Wire:
- Wire carries uniform current I_0.
- Need to find magnetic field produced by this current everywhere in space.
Distance and Current Element:
- Choose a point at distance r from the wire where we will evaluate the magnetic field.
- Consider a small element of wire with distance dr carrying a small current.
Magnetic Field Equation:
- Magnetic field B due to a current element can be expressed as:
  B = \frac{\mu0}{4\pi} \int \frac{I0 \, d\textbf{l} \times \hat{r}}{r^2}
- Here, \hat{r} is the unit vector pointing towards the evaluation point.
Use of Cross Product:
- Need to evaluate d\textbf{l} \times \hat{r}, where d\textbf{l} = d\textbf{y} (infinitesimal distance in y-direction).
Coordinate Choice:
- Choose coordinates where y is vertical and x is horizontal.
- The angle between current element d\textbf{l} and \hat{r} is denoted as \phi.
Sine of Angles:
- By geometry, \sin(\phi) can be expressed in terms of coordinates: \sin(\phi) = \frac{x}{\sqrt{x^2 + y^2}}.
- Using right-hand rule for direction, B points in negative k direction (
  -\hat{k}).
Final Expression of Magnetic Field:
- The magnetic field expression simplifies to:
  B = \frac{\mu0}{4\pi} \int \frac{I0 \, x}{(x^2 + y^2)^{3/2}} \, dy \, (-\hat{k}).
Integration Limits:
- Integrate from y = -\infty to y = +\infty to find the entire field due to all current elements along the wire.