Step 1: Magnetic Field Distribution Determination Ampère's Circuital Law dictates the magnetic field (\( B \)) around a current-carrying wire: \[ \oint B \cdot dl = \mu_0 I_{\text{enc}} \] \( I_{\text{enc}} \) represents the current encompassed by the selected circular Amperian loop.
Step 2: Magnetic Field Within the Wire (\( r<a \)) Assuming uniform current distribution, the enclosed current at radius \( r \) (\( r<a \)) is calculated as: \[ I_{\text{enc}} = I \frac{\pi r^2}{\pi a^2} = I \frac{r^2}{a^2} \] Applying Ampère’s law to a circular path of radius \( r \): \[ B \cdot 2\pi r = \mu_0 I_{\text{enc}} \] Substituting \( I_{\text{enc}} \): \[ B \cdot 2\pi r = \mu_0 I \frac{r^2}{a^2} \] \[ B = \frac{\mu_0 I r}{2\pi a^2} \] This result establishes a
direct proportionality between the magnetic field within the wire and \( r \).
Step 3: Magnetic Field External to the Wire (\( r>a \)) For radii \( r>a \), the Amperian loop encloses the total current \( I \): \[ B \cdot 2\pi r = \mu_0 I \] \[ B = \frac{\mu_0 I}{2\pi r} \] Consequently, the magnetic field outside the wire exhibits an
inverse proportionality to \( r \).
Final Answer: The magnetic field is directly proportional to \( r \) inside the wire and inversely proportional to \( r \) outside the wire.