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| ### Statement |
| ### Statement |
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| $9.3.15.$ [Insert the problem statement] |
| $9.3.15.$ The circulation of the induction of a constant magnetic field through a closed circuit in a vacuum is equal |
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| to the current through the surface bounded by this circuit, multiplied by $µ0$. Please provide examples |
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| that support this law. Use it to solve the following problems: |
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| a. An infinitely long straight wire of radius $r$ flows current $I$. The current is distributed evenly over the |
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| wire cross-section. Find the magnetic field induction inside and outside the wire. |
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| b. A long, wide busbar with a cross dimension of a flows a current evenly distributed over the crosssection of the conductor. Current density $j$. How does the magnetic field induction depend on the distance $x$ to the median plane of the tire? |
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| ### Solution |
| ### Solution |
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| а) Возьмем круглый контур радиуса $R$ при $R>r$ весь ток входит в контур: |
| а) |
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| Let's take a circular contour of radius $R$, when $R>r$ all the current enters the contour: |
| $$B\cdot 2\pi R = \mu_0 I$$ |
| $$B\cdot 2\pi R = \mu_0 I$$ |
| $$B=\frac{\mu_0 I}{2\pi R}$$ |
| $$B=\frac{\mu_0 I}{2\pi R}$$ |
| При $R < r$： |
| with $R < r$： |
| $$B\cdot 2\pi R= \mu_0 I \frac{R^2}{r^2}$$ |
| $$B\cdot 2\pi R= \mu_0 I \frac{R^2}{r^2}$$ |
| $$B=\frac{\mu_0 I R}{2\pi r^2}$$ |
| $$B=\frac{\mu_0 I R}{2\pi r^2}$$ |
| б) Рассмотрим прямоугольный контур размерами $L$ на $2x$, при $x < a/2$: |
| b) Let's consider a rectangular contour with dimensions |
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| $L$ and $2x$, with $x < a/2$: |
| $$B \cdot 2L = \mu_0 \cdot 2jxL$$ |
| $$B \cdot 2L = \mu_0 \cdot 2jxL$$ |
| $$B = \mu_0 jx$$ |
| $$B = \mu_0 jx$$ |
| При $x >= a/2$: |
| with $x >= a/2$: |
| $$B = \frac{\mu_0ja}{2}$$ |
| $$B = \frac{\mu_0ja}{2}$$ |
| У Савченко опечатка в знаке неравенства |
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| Savchenko has a typo in the inequality sign |
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| #### Answer |
| #### Answer |
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| [Insert a concise answer or boxed result] |
| а. $B = \frac{\mu_0Ix}{2πr^2}$, $0 < x < r$; $B = \frac{\mu_0I}{2πx}$, $x > r$. |
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| b. $B =\mu_0 xj$, $x < a/2$; $B =\frac{\mu_0 aj}{2}$, $x >= a/2$. |
