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:
  $B = \frac{\mu<em>0}{4\pi} \int \frac{I</em>0 \, d\textbf{l} \times \hat{r}}{r^2}$
- 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:
  $B = \frac{\mu<em>0}{4\pi} \int \frac{I</em>0 \, x}{(x^2 + y^2)^{3/2}} \, dy \, (-\hat{k})$ .
Integration Limits:
- Integrate from $y = -\infty$ to $y = +\infty$ to find the entire field due to all current elements along the wire.