Electromagnetic Fields: Gradient

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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Vectorial operators/Gradient

1.6 Vectorial Operators

In the use of concepts in electromagnetic theory, the use of vector operators is necessary. They allow us to mathematically express complicated models in a simple and independent way of the system of coordinates used.

To interpret and adequately use the Maxwell equations, it is necessary to understand the vector operations that we will describe in this section. On another hand, the solution of practical problems requires the knowledge to apply these vector operators in different coordinate systems.

1.6.1 Gradient.

This first operator has as an argument a scalar function and produces as a result a vector function. The gradient of the scalar function “V”, is written the following way,

(34)

The before mentioned vector has as a direction, that in which the scalar function “V” varies the fastest and as a magnitude, the derivative of function “V” in the above direction.

For example, close a light bulb, supposing there were no other heat sources in the region, the gradient of temperature in any point would be a vector that would be pointing towards the light bulb.

The vector field constituted by the gradient of a scalar function “V”, will indicate in each point, how and to where function “V” is varying. We can determine the differential variation of function “V” in any direction with the following scalar product.

(35)

When both multiplicands are parallel, this expression is maximum; when the multiplicands are perpendicular, the result is zero; and we will then say that dl is on an equipotential surface.

The variation of function “V” between points “a” and “b”, could be determined through a line integral,

        (36)

The gradient in the systems of coordinates that we have described before will be

Rectangular:

        (37)

Cylindrical:

      (38)

Spherical:

        (39)


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