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Appendices
Appendix II: Vector Algebra
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Vector algebra is formulated to handle vectors, that is, quantities with both magnitude and direction. Normal algebra, geometry, and trigonometry are efficient at dealing with scalar quantities, that is, those with only magnitude, but are inefficient at handling vectors. Vector algebra is an efficient way of solving 2D and 3D problems involving vectors without the need for cumbersome geometry.
The electromagnetic field is a vector field of forces acting on charged objects. As forces are vector quantities, the EM field equations involve vectors.
Vector algebra can be formulated for Cartesian, cylindrical, and spherical coordinates. Appropriate choice of coordinate system for cylindrically or spherically symmetric problems avoids needless complexity arising from the use of an inappropriate coordinate system and will also clearly show the symmetry of the solution.
Two important results of vector algebra involve multiplication of vectors. (Vectors are shown in bold text.)
The dot product A.B (called A dot B) is defined as AB cos θ, where A and B are the magnitudes of the vectors A and B respectively, and θ is the smaller angle between them. Note that the dot product is a scalar quantity.
The cross product AxB ("A cross B") is defined as (AB sin θ) an where an is the unit vector normal to the plane of A and B. Note that the cross product of two vectors is also a vector and is orthogonal to both A and B, that is, the resultant vector involves a third dimension compared to the plane of the first two.
In addition, vector algebra defines an important operator, Del, or ∇. Del is analogous to the differential operator D in calculus, where D represents the operation d/dx. Two further results using Del are important in analyzing EM fields.
∇A, called Div A, is the divergence of the vector field A. The divergence is a scalar and similar to the derivative of a function. If the divergence of a region of a vector field is nonzero, then that region is said to contain sources (Div > 0) or sinks (Div < 0) of the field. For example, a closed surface around an isolated positive point-charge in a static electric field contains the source of the electric flux, that is, the positive point-charge, therefore the divergence of the electric flux density vector field over that surface will be positive and equal to the enclosed charge. This is Gauss's Law.
Note that Div A involves a dot product and is therefore dependent on angles. The angle is usually that between the vector and the normal to the surface being analyzed.
∇xA, called Curl A, is the curl of the vector field AA. The curl of a vector field is another vector field which describes the rotation of the first vector field: the magnitude of ∇xA is the magnitude of the rotation; the direction of ∇xA is the axis of that rotation as determined by the right-hand rule. If one imagines any 3D vector field to represent fluid flow velocities, then the curl of the field at a point would be indicated by the way a small sphere placed at that point would be rotated by the flow. In a 2D flow it is easy to see that the direction of the axis will be in the 3rd dimension, as is given by the use of the cross product in calculating the curl.
Additionally, the Del operator can be applied to a scalar field V. ∇V or Grad V is a vector field defining the gradient of the scalar function V. ∇V lies in the direction of the maximum increase of the function V. If applied to a potential function, then Grad V is a vector field that is everywhere normal to the equipotential surfaces.
Two useful properties of the Curl operator are:
- The divergence of the curl of any vector field is the zero scalar; that is, ∇.(∇xA) = 0
- The curl of a gradient of any vector field is the zero vector, that is, ∇x(∇f) = 0 for any scalar function of position, f.
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end of Appendix 2
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next: Appendix III: The Electro-magnetic Field Equations
previous: Appendix I: The Nature of Electro-magnetic Fields
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