26.10
连续性方程
单位横截面积上流过的电流总量称为电流密度。 因此，通过横截面积的电流可以写为电流密度的表面积分。
根据电荷守恒定律，流出给定体积的总电流等于该体积内电荷的减少率。
现在，总电荷可以用体积电荷密度来写。
电荷密度是空间函数。 因此，应用莱布尼兹规则，时间导数可以移动到积分内部。
当将散度定理应用于上式…
电流密度是单位横截面积上流过的总电流。因此，通过某一横截面积的总电流可表示为电流密度的面积分。
现在，根据电荷守恒定律，流出某一给定体积的总电流必须等于该体积内电荷量的减少速率。
表达 根据体电荷密度表示总电荷时，该方程需作相应修改。由于电荷密度是空间函数，利用莱布尼茨法则，可将其对时间的导数移入积分号内。
将散度定理应用于该方程的左侧，可将闭合曲面积分转化为体积分。此关系适用于任意体积；因此，被积函数相等。
电流密度的散度等于体电荷密度变化率的负值。这一表示电荷局域守恒的数学表述称为连续性方程。
对于稳恒电流，电荷密度不随时间变化。因此，电流密度的散度为零。
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Q1: What is current density and how does it relate to total current?
Current density is the total amount of current flowing per unit cross-sectional area. The total current passing through a cross-sectional area can be expressed as the surface integral of the current density. This relationship allows us to calculate total current from the distribution of current density across a surface.
Q2: How does charge conservation relate to current flow in a volume?
Charge conservation requires that the total current flowing out of a given volume equals the rate of decrease of charge within that volume. This principle connects the outward flow of electrical charge to changes in the charge stored inside the volume, establishing a fundamental relationship between current and charge density.
Q3: What mathematical steps lead to the continuity equation?
Starting with charge conservation, the total charge is expressed in terms of volume charge density. Using the Leibniz rule, the time derivative moves inside the integral. Applying the divergence theorem converts the closed surface integral to a volume integral. Since this holds for any volume, the integrands must be equal, yielding the continuity equation.
Q4: What does the continuity equation state about current density and charge density?
The continuity equation states that the divergence of the current density equals the negative rate of change of volume charge density. This mathematical statement represents local charge conservation, showing how spatial variations in current relate to temporal changes in charge density at any point.
Q5: Why is the divergence of current density zero for steady currents?
For steady currents, the charge density is invariant with time, meaning it does not change. Since the continuity equation states that divergence of current density equals the negative rate of change of charge density, and this rate is zero for steady conditions, the divergence of current density must also be zero.
Q6: How does the divergence theorem simplify the continuity equation derivation?
The divergence theorem converts the closed surface integral on the left side of the charge conservation equation into a volume integral. This transformation allows both sides of the equation to be expressed as volume integrals, enabling direct comparison of integrands and leading to the local form of the continuity equation.
Q7: What role does volume charge density play in the continuity equation?
Volume charge density represents the charge per unit volume as a space function. Since it varies with position, the Leibniz rule allows the time derivative to move inside the integral during derivation. The continuity equation directly relates changes in volume charge density to the divergence of current density at each point in space.