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naz edited Jun 11, 2026, 5:24 PM

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revision #18622 newer →

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		+	### Statement
		+
		+	$9.4.3.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	We know that a plane with linear current density $i$ creates a uniform field around itself, its value $$B=\frac{\mu_0 i}{2}$$
		+	, but we don't really need it.
		+
		+	Consider an arbitrary closed surface. The stream is by definition equal to:
		+	$$\oint_S\hat{B}\cdot d\hat{S}$$
		+
		+	The definition of vector divergence will be useful to us.:
		+	$$div \hat{a} = {lim}_{\Delta V\to0} \frac{1}{\Delta V}\oint_S \hat{a}\cdot d\hat{S}=\frac{\partial a}{\partial x}+\frac{\partial a}{\partial y} + \frac{\partial a}{\partial z}$$
		+
		+	Note for now that the second definition and uniformity of the field implies $$div\hat{B}=0$$
		+	in the whole space
		+
		+	Now let's break down the volume that borders our surface into many infinitely small volumes $dV$. From the first definition of divergence, an alternative expression for the flow can be obtained:
		+	$$\Phi = \int_V div \hat{B} dV$$
		+
		+	Because the divergence is zero at any point in space.: $\to$ $$\Phi=0$$
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$9.4.3.$ [Insert the problem statement]

			### Solution

			We know that a plane with linear current density $i$ creates a uniform field around itself, its value $$B=\frac{\mu_0 i}{2}$$
			, but we don't really need it.

			Consider an arbitrary closed surface. The stream is by definition equal to:
			$$\oint_S\hat{B}\cdot d\hat{S}$$

			The definition of vector divergence will be useful to us.:
			$$div \hat{a} = {lim}_{\Delta V\to0} \frac{1}{\Delta V}\oint_S \hat{a}\cdot d\hat{S}=\frac{\partial a}{\partial x}+\frac{\partial a}{\partial y} + \frac{\partial a}{\partial z}$$

			Note for now that the second definition and uniformity of the field implies $$div\hat{B}=0$$
			in the whole space

			Now let's break down the volume that borders our surface into many infinitely small volumes $dV$. From the first definition of divergence, an alternative expression for the flow can be obtained:
			$$\Phi = \int_V div \hat{B} dV$$

			Because the divergence is zero at any point in space.: $\to$ $$\Phi=0$$

			#### Answer

			[Insert a concise answer or boxed result]