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The Gradient of a Scalar Field

Fundamentals of Engineering Electromagnetics, by David Cheng, defines the gradient of a scalar field as follows:

    We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar

But what does that mean, and why is it useful?

First, we need to understand the concept of a scalar field. In three dimensions, a scalar field is simply a field that takes on a sinlge scalar value at each point in space. For example, the temperature of all points in a room at a particular time t is a scalar field.

The gradient of this field would then be a vector that pointed in the direction of greatest temparature increase. Its magnitude represents the magnitude of that increase.

To calculate the gradient of a vector field in Cartesian coordinates, the following method is used :

Given : S is a scalar field ( S is some function of x , y , and z)
Find : grad S
                                              _            _             _
        grad S = del S , del= x(d/dx) + y(d/dy) + z(d/dz)

        Thus :
                         d      _        d      _         d     _
        grad S = -----S x + ------S y + ------S z
                        dx              dy               dz


If you are working in other coordinates systems, remember that the del operator is defined in terms of length. Since methods of representing physical lengths differ from system to system, be sure you use the correct representation for your problem.

Now for the question of why this is useful in electromagnetics. One of the properties of a conservative vector field (such as the electrostatic field) is that it can be expressed as the gradient of a scalar field . The intensity of an electrostatic field, for example, is related to the gradient of a scalar field that we call voltage. In many cases, since the voltage is a number and not a vector, it is easier to solve a problem for voltage. Knowing how to express the gradient of this field allows us to calculate E in a simpler manner. E=-(grad V)

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