# Physical intepretation of derivative in Maxwell equation?

• DunWorry
In summary, Maxwell's equations show that a changing magnetic flux in time produces a potential difference and electric field in space. This is represented by a full derivative with respect to time. The curl of the electric field at a certain point also produces a changing magnetic field at that point. The equations work both ways, and B includes spatial characteristics while the dt derivative shows how it changes with time.

#### DunWorry

I'm having a little difficulty understanding the use of derivatives in Maxwell's equations. Eg. $\oint E . dl$ = - $\frac{d\varphi_{B}}{dt}$ this says that a changing magnetic flux in time, produces a potential difference (and electric field) in space? I noticed that its a full derivative, and its dt. Whats the significance of this? why would it be wrong if it was magnetic flux changing in space or something?

This can be re-written as $\nabla x E$ = - $\frac{dB}{dt}$ So a curling electric field in space, produces a changing magnetic field that varies in time? how come there is no space dependence on the magnetic field? like in an EM wave the magnetic field doesn't just stay in one spot and change its magnitude, it propagates with the electric field.

Perhaps its my understanding of curl? or does it mean the curl of the electric field at a certain point in the field, produces a changing magnetic field at that point also?

I'm not sure =D
Thanks!

You have it backwards. It's not that "a curling electric field in space, produces a changing magnetic field that varies in time", it's that a time-varying magnetic field produces an electric field with non-zero curl. The right-hand side is the source of the field on the left-hand side.

The equations work both ways.

B represents a vector magnitude and direction, so it's spatial characteristics are included. The dt derivitive indicates how it changes with time.

## 1. What is the physical meaning of the derivative in Maxwell's equations?

The derivative in Maxwell's equations represents the rate of change of a physical quantity with respect to another physical quantity. It can be interpreted as the slope of a curve, or the instantaneous change of a physical quantity at a specific point in space and time.

## 2. How does the derivative relate to the electric and magnetic fields in Maxwell's equations?

The derivative of the electric field with respect to time is related to the rate of change of the magnetic field, while the derivative of the magnetic field with respect to time is related to the rate of change of the electric field. This relationship is described by Faraday's and Ampere's laws respectively, which are two of Maxwell's equations.

## 3. What is the significance of the second derivative in Maxwell's equations?

The second derivative in Maxwell's equations represents the curvature or acceleration of a physical quantity. For example, the second derivative of the electric field with respect to time describes the acceleration of the electric field, which is related to the changing magnetic field.

## 4. Can the derivative in Maxwell's equations be negative?

Yes, the derivative in Maxwell's equations can be negative. A negative derivative represents a decreasing rate of change, or a downward slope on a curve. This can occur when the magnetic field is decreasing in strength, or when the electric field is decreasing in time.

## 5. How is the derivative used in the physical interpretation of Maxwell's equations?

The derivative is used to describe the behavior and interactions of electric and magnetic fields in space and time. It helps to explain phenomena such as electromagnetic radiation and how electric and magnetic fields can induce each other. The derivative also allows for the prediction and calculation of these phenomena, making it a crucial tool in understanding and applying Maxwell's equations.