Question 1. Find the magnetic field strength at the center of curvature of a half-circle of thin wire in the positive-y half of the xy-plane with a radius of half a meter carrying a uniform current of 2 mA counterclockwise around the z-axis if the permeability is 4πx10^-7 T·m/A. 2. A straight wire carries current I along the z-axis from -z0 to +z0 in a region of
1. Find the magnetic field strength at the center of curvature of a half-circle of thin wire in the positive-y half of the xy-plane with a radius of half a meter carrying a uniform current of 2 mA counterclockwise around the z-axis if the permeability is 4πx10^-7 T·m/A.
2. A straight wire carries current I along the z-axis from -z0 to +z0 in a region of permeability µ. Find the magnetic field vector at x=x0 on the x-axis.
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Using Biot-Sqvert kus\begin{aligned}\overrightarrow{d B} & =\frac{\mu_{0} I}{4 \pi r^{3}}(\vec{r} \times \overrightarrow{d l}) \\d B & =\frac{\mu_{0} I}{4 \pi r^{3}} r \cdot d l \sin 90^{\circ} \\& =\frac{\mu_{0} I}{4 \pi r^{3}} r \cdot r d \theta . \quad \because d l=r \cdot d \theta \\B & \pi\end{aligned}\begin{aligned}\int_{0}^{B} d B & =\frac{\mu_{0} I}{4 \pi} \int_{0}^{\pi} d \theta \\B & =\frac{\mu_{0} I}{4 r} \\B & =\frac{\pi \times 10^{-7} \times 2 \times 10^{-3}}{0.5} \\& =12.57 \times 10^{-10}+(-\hat{k})\end{aligned}9 \operatorname{long}-2B=\frac{\mu_{0} I}{4 \pi x_{0}}\left(\sin \theta_{1}-\sin \theta_{2}\right) \quad \theta_{1}=-\theta_{2}\sin \theta_{2}=\sin \theta_{1}=\frac{20}{\sqrt{20^{2}+x_{0}^{2}}}\begin{aligned} B & =\frac{\mu_{0} I}{4 \pi x_{0}} 2 \sin \theta_{1} \\ & =f^{2} \\ B & =\frac{\mu_{0} I}{2 \pi x_{0}} \frac{z_{0}}{z_{0}^{2}+x_{0}^{2}} \quad \text { along } \quad y\end{aligned} ...