Why partial derivatives in continuity equation?

Thread starter biubiu
Start date Jun 14, 2012
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Continuity Continuity equation Derivatives Partial Partial derivatives

AI Thread Summary

The continuity equation employs a partial derivative with respect to time to account for the spatial variation of charge density, represented by ρ. This approach is necessary because the charge density can change not only over time but also across different spatial locations. The equation, ∂ρ/∂t = -∇·j, reflects the local rate of change of charge density due to the flow of charge, indicated by the current density vector j. While the total derivative dq/dt considers the overall change in charge, the partial derivative is more appropriate for analyzing localized changes in a field. Understanding this distinction is crucial for accurately modeling physical systems in fluid dynamics and electromagnetism.

Jun 14, 2012

biubiu

Why is partial derivative with respect to time used in the continuity equation,
<br /> \frac{\partial \rho}{\partial t} = - \nabla \vec{j} <br />
If this equation is really derived from the equation,
<br /> \frac{dq}{dt} = - \int\int \vec{j} \cdot d\vec{a}<br />
Then should it be a total derivative with respect to time?

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Jun 14, 2012

vin300

Partial derivative is used because the charge density may also vary with distance.

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