A wire of radius R carries a current I.The current density is given by the function J = J₀ (1 - r/R)where r is measured from the center of the wire and J₀ is a constant.Use Ampère's law to find the magnetic field inside the wire at a distance r

Question

A wire of radius R carries a current I.The current density is given by the function J = J₀ (1 - r/R)where r is measured from the center of the wire and J₀ is a constant.Use Ampère's law to find the magnetic field inside the wire at a distance r < R from the axis.What is the magnitude of B if I = 1A,R = 1.5 mm,and r = 1 mm? A)12 mT B)37 µT C)1.5 T D)45 mT E)148 µT

Quizplus · Accepted Answer

To find the magnetic field inside the wire, we use Ampère's law in the form $B \cdot 2\pi r = \mu_0 I_{\text{enc}}$, where $I_{\text{enc}}$ is the current enclosed by the loop of radius $r$. Given $J = J_0(1 - \frac{r}{R})$, the enclosed current is found by integrating the current density over the cross-sectional area up to radius $r$, leading to $I_{\text{enc}} = \int_0^r J_0(1 - \frac{r'}{R}) 2\pi r' dr'$. Solving this gives $I_{\text{enc}} = \frac{\mu_0 I r^2}{2R^2}$, and substituting the given values yields the magnetic field magnitude.

A Wire of Radius R Carries a Current I