EM_Equations-RevC

EM_Equations-RevC - ECE 3025C Electromagnetics Orthogonal...

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ECE 3025C -- Electromagnetics Useful Equations For E & M September 2007 Orthogonal Coordinate Systems Consider 12 3 x , x , and x to be three axis labels for a coordinate system, and further assume that these axes are orthogonal, or at least locally orthogonal. Let 3 , , and ee e ±± ± be unit vectors along the three respective axes. A differential length vector ds JJG , with coordinate components () 123 ,, ds ds ds ds = , is introduced to preserve differential length changes for various coordinate systems. Consider the coordinate change ( ) dx dx dx dx = in a general coordinate system. This change is a mixture of angular and linear coordinates for systems other that Cartesian. Thus, we introduce ( ) 1 1 2 2 3 3 , , ds ds ds ds h dx h dx h dx == , where 3 h , h , and h are factors to convert angular coordinates to true length units. Using this notation, the following relations are proper for the cartesian, cylindrical, and spherical coordinate systems. Cartesian Cylindrical Spherical Relationship to Cartesian N/A cos sin xr yr zz θ = = = sin cos sin sin cos zr φ = = = Parameter Cartesian Cylindrical Spherical 1 x x r r 2 x y 3 x z z 1 h 1 1 1 2 h 1 r r 3 h 1 1 sin r 11 1 ds h dx = dx dr dr 22 2 ds h dx = dy rd 33 3 ds h dx = dz dz sin Elemental volume change dV ds ds ds = dx dy dz rdrd dz 2 r d d θϕ
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ECE 3025C -- Electromagnetics Useful Equations For E & M Page 2 of 17 Indicial Notation Indicial notation is often used for mathematical expressions involving vectors, tensors, and matrices. Specifically, a repeated index on the right hand side of an equation implies a sum is to be made over the repeated indices. To illustrate indicial notation, consider C HG , a (3 x 3) matrix, and B JG a (3 x 1) vector. The matrix product C B A = H GJ G J G i is a (3 x 1) matrix, and may be expressed in either of the following forms: 1 11 12 13 21 22 23 2 31 32 33 3 ii j j j B CCC A C C C B B C B ⎛⎞ ⎜⎟ == ⎝⎠ = i In indicial notation this matrix product is written as: j j = The repeated index j on the right hand side of this equation implies that the sum is to be made over j , and the equation states that the th i element of A is given by: () j j i j j i 1 1 i 2 2 i 3 3 j A CB CB C B CB = + + Indicial notation is especially useful for efficiently writing dot, cross, and matrix product terms in tensor notation, and is also referred to as "summation" notation. The following tensor discussion illustrates several examples of indicial notation. Tensors Scalars – 0 th Rank Tensors Many physical parameters and measurables may be represent in space by a scalar parameter, such as temperature, density, voltage potential. Scalars do not depend upon the direction in space, and thus may be expressed as a single parameter, i.e., the measurement value does not depend upon direction.
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ECE 3025C -- Electromagnetics Useful Equations For E & M Page 3 of 17 Vectors – 1 st Rank Tensors All vectors are referred to as 1 st rank tensors. Force, velocity, acceleration, electric field, and magnetic field intensity are all examples of vector quantities. Measurement values of vector
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EM_Equations-RevC - ECE 3025C Electromagnetics Orthogonal...

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