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Problem 48 Hard Difficulty

Experiments show that a steady current $I$ in a long wire produces a magnetic field $\mathbf{B}$ that is tangent to any circle that lies in the plane perpendicular to the wire and whose
center is the axis of the wire (as in the figure at the right). Ampere's Law relates the electric current to its magnetic effects and states that
$$ \int_{C} \mathbf{B} \cdot d \mathbf{r}=\mu_{0} I$$
where $I$ is the net current that passes through any surface bounded by a closed curve $C,$ and $\mu_{0}$ is a constant called the permeability of free space. By taking $C$ to be a circle
with radius $r,$ show that the magnitude $B=|\mathbf{B}|$ of the magnetic field at a distance $r$ from the center of the wire is $$B=\frac{\mu_{0} I}{2 \pi r}$$

Answer

$\int_{C} \mathbf{B} \cdot d \mathbf{r}=2 \pi r B=\mu_{0} I$
$B=\frac{\mu_{0} I}{2 \pi r}$

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Video Transcript

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