26.10
Équation de continuité
La quantité totale de courant traversant par unité de surface transversale est appelée densité de courant. Ainsi, le courant traversant une surface tr…
La densité de courant est la quantité totale de courant circulant par unité de section transversale. Ainsi, le courant total passant par une section transversale peut être exprimé comme l’intégrale de surface de la densité de courant.
Or, la conservation de la charge exige que le courant total sortant d’un volume donné soit égal au taux de diminution de la charge à l’intérieur de ce volume.
En exprimant la charge totale en termes de densité de charge volumique, l’équation est modifiée. Comme la densité de charge est une fonction spatiale, en utilisant la règle de Leibniz, sa dérivée temporelle peut être déplacée à l’intérieur de l’intégrale.
L’application du théorème de divergence au membre gauche de l’équation convertit l’intégrale de surface fermée en intégrale de volume. Cette relation est valable pour tout volume ; En conséquence, les intgrands sont égaux.
La divergence de la densité de courant est égale au taux de variation négatif de la densité de charge volumique. Cette affirmation mathématique de la conservation de la charge locale s’appelle l’équation de continuité.
Pour les courants stationnaires, la densité de charge est invariante avec le temps. Ainsi, la divergence de la densité de courant est nulle.
View the full transcript and gain access to JoVE Core videos
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.