Displacement Current Density - The Partial Derivative of D

Maxwell's Equations (Home)

Partial Derivative in Time of the Electric Flux Density

On this page, we'll give the meaning of the following term in Ampere's Law:

the symbol for the partial derivative of the electric flux density
[Equation 1]

In short, this is simply the time rate of change of the Electric Flux Density. That is, this quantity is a measure of how quickly the D field changes if we observe it as a function of time. This is different than if we look at how the D field changes spatially - i.e. over a region of space for a fixed amount of time.

This term is known as the Displacement Current Density. It symbolizes the type of electric current that flows through a capacitor. Displacement current is different than the current that flows through a wire or an inductor - this is carried by free electrons on a conductor, and is known as conductive current. We know that in some sense current flows through a capacitor, but there is no physical conductive connection between the parallel plates of a capacitor. And to ensure current flowing in a circuit or loop remains constant, we need this term. This was actually introduced to Ampere's Law by Mr. Maxwell himself, which was a stroke of genius that laid the theoretical basis for wave propagation, amongst other things (See Ampere's Law).

To discuss this more on math terms, suppose we have an electric flux density D, which is a vector field and a function of (x,y,z,t) (3-spatial variables and time). We can break the D field down into its x-, y- and z-components:

the electric flux density definition, in terms of it's spatial components
[Equation 2]

If you recall the partial derivative page, we know that the partial derivative of D with respect to time is the rate of change of the D field in time (that is, we ignore any spatial variation in the D field and are only concerned with how it changes versus time). Hence, we can rewrite Equation [1] as:

electric flux density partial derivative in time
[Equation 3]

If you understand the Electric Flux Density page and the Partial Derivatives page, Equation [3] should be pretty clear. If not, run through those topics and you'll see this isn't very complicated.

To conclude, since the D field is measured in Coulombs/meter^2 [WCm^2] and the derivative means "per second" [1/s], then the units of Equation [1] are Coulombs/meter-squared-seconds [C/m^2-s]. But we also know from the Ampere's Law Equality Page that this is equivalent to Amps/meter^2 [A/m^2].

Maxwell's Equations

This page on the time-partial derivative of the electric flux density or displacement current density has been copyrighted. The copyright belongs to Maxwells-Equations.com, 2012.