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Transcript
ELECTROMAGNETIC FIELD WAVE LECTURE-2 Vector Analysis The quantities that we deal in electromagnetic theory may be either scalar or vectors [There are other classes of physical quantities called Tensors: scalars and vectors are special cases]. Scalars are quantities characterized by magnitude only and algebraic sign. A quantity that has direction as well as magnitude is called a vector. Both scalar and vector quantities are function of time and position. A field is a function that specifies a particular quantity everywhere in a region. Depending upon the nature of the quantity under consideration, the field may be a vector or a scalar field. Example of scalar field is the electric potential in a region while electric or magnetic fields at any point is the example of vector field. Fundamentals Vector Algebra: A vector is represented by a directed line segment: length of the line is proportional to magnitude and the orientation of the directed line segment with respect to some reference gives the vector. ˆ , where A A is the magnitude and aˆ A vector A can be written as A aA vector which has unit magnitude and direction same as that of A . Two vectors A and B are added together to give another vector C . We have: C A B Figure 1: Vector addition Vector subtraction is similarly carried out as: D A B A ( B ) Prepared by A.Das A is the unit A ELECTROMAGNETIC FIELD WAVE Figure 2: Vector subtraction Scaling of a vector is defined as C B , where C is a scaled version of vector B and is a scalar. Some important laws of vector algebra are: A B B A A ( B C ) ( A B) C Commutative law Associative law ( A B) A B Distributive law The position vector rP of a point P is the directed distance from the origin ( O ) to P i.e. rP OP . Figure 3: Distant vector If rP OP and rQ OQ are the position vectors of the points P and Q then the distance vector PQ OQ OP rQ rP Product of Vectors When two vectors A and B are multiplied, the result is either a scalar or a vector depending how the two vectors were multiplied. The two types of vector multiplication are: Scalar product (or dot product) A B gives a scalar Vector product (or cross product) A B gives a vector Prepared by A.Das ELECTROMAGNETIC FIELD WAVE The dot product between two vectors is defined as A B AB cos AB . Figure: Dot product The dot product is commutative i.e. A B B A and distributive i.e. A ( B C ) A B A C . Associative law does not apply to scalar product. The vector or cross product of two vectors A and B is denoted by A B . A B is a vector perpendicular to the plane containing A and B , the magnitude is given by ABSin AB and direction is given by right hand rule as explained in Figure. Figure: Vector cross product A B aˆn ABSin AB , where aˆn aˆ n is the unit vector given by A B A B The following relations hold for vector product. A B B A i.e. cross product is non commutative A (B C) A B A C i.e. cross product is distributive A ( B C ) ( A B) C i.e. cross product is non associative Scalar and vector triple product Scalar triple product Prepared by A.Das A ( B C ) B (C A) C ( A B) ELECTROMAGNETIC FIELD WAVE Vector triple product A ( B C ) B( A C ) C ( A B) Prepared by A.Das