The Partial Derivative of B

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The Partial Derivative of the Magnetic Flux Density

On this page we define the symbol:

the symbol for the partial derivative of a function
[Equation 1]

Briefly, this is simply the negative of the rate of change of the B field with respect to time. The Magnetic Flux Density (B) is defined here.

To define this fully, let's say we have a magnetic flux density B, which is a vector field and a function of (x,y,z,t) (3-spatial variables and time). We write the B field in terms of its x-, y- and z-components:

the magnetic flux density definition
[Equation 2]

From the partial derivative page, we know that the partial derivative of B with respect to time is the rate of change of the B field in time (that is, we ignore any spatial variation in the B field and are only concerned with how it changes versus time). And the negative sign in Equation [2] simply negates each of the components. Hence, we can rewrite Equation [1] as:

magnetic flux density partial derivative in time
[Equation 3]

If you understand the Magnetic Flux Density page and the Partial Derivatives page, Equation [3] follows pretty naturally. If not, check out those topics.

Finally, since the B field is measured in Webers/meter^2 [Wb/m^2] and the derivative means "per second" [1/s], then the units of Equation [1] are Webers/meter-squared-seconds [Wb/m^2-s]. But we also know from the Faraday's Law Equality Page that this is equivalent to Volts/meter^2 [V/m^2].

Maxwell's Equations

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