1. مفردات منهج الكهرومغناطيسية CH 1: vector analysis Change in Cartesian coordinates systems. Change of axis (Rotation Matrices). Field and differential operators. Derivative and integration operators Cylindrical and spherical coordinates. Divergence theorem. Stoke theorem. CH 2: Electrostatics Coulomb’s law. Gauss’s law. Poisson’s equation. Solution of Laplace’s equation in : Cylindrical coordinates. Spherical coordinate. Rectangular coordinate. 1

2. CH 3: Electrostatic field in dielectric medium Potential and electric field outside a dielectric medium ( polarization ). Electric field inside a dielectric. Gauss’s law in dielectric ( electric displacement ). Properties of dielectric. Poisson’s and Laplace’s equation in dielectric. Dielectric sphere in uniform field. Microscopic theory of the dielectric. CH 4: Electric energy The energy in electric field. The energy in dielectric. Capacitors. CH 5: conduction Electric current. Current density. Equation of continuity. Ohm’s law. Steady current in a conductor. 2

3. CH 6: Magnetostatics Biot and Savart law. Force on a current element in magnetic flux B. Ampere’s law. CH 7: Electromagnetic induction Faraday’s law in induction. Inductance. The LCR circuit CH 8: Magnetization Boundary condition of field vector. Microscopic theory of magnetic properties of matter. Magnetic energy and energy density. Maxwell equation. Maxwell equation and their application. 3

4. 1-Vector in Cartesian coordinate systems :- Roughly, a vector is a quantity that has magnitude and direction and that behave like the position vector under addition and multiplication and under rotation. Position vector: the position vector of a point with given coordinates x,y,z is the three – component object. superscript k has the value k=1,2,3 called row vector Y X Z P z y x 4

5. Addition of position vector is defined as addition of their component for exp. (5,3,0)+(-1,0,2)=(4,3,2) Multiplication of vector by number is defined as multiplication of each component. Exp. 2(-1,0,2)=(-2,0,4) The unit vector along the rectangular axis are define as : (1,0,0), (0,1,0), (0,0,1) We can express an arbitrary vector u as a superposition of unit vector. a- Length ( magnitude ) of vector X y u U,j U,i 5

6. x y Z U,K U,i U,j U c- Scalar product ( dot product ) of two vector : Sol. Ex. Find the angle between the vectors and, A B θ b- Projection of vector on coordinate axis : 6

7. H.W : Find the projection of the vector on ? Sol. H.W : Find the angle between the vectors A and the axis's (x,y,z) ? A=3i-6j+2k A x =A. i =|A| |i| cosθ x A y =A. j =|A| |j| cosθ y A z =A.K=|A| |K| cosθ z A B θ C O K i j Z X Y ΘxΘx ΘyΘy ΘZΘZ A 7

8. d- Vector product ( cross product ) : 1- 2- 3- 4- H.W:, prove that ? Ex. Find the shortest distance between the point (6,-4,4) and the line connected between the two point (3,-1,4),(2, 1,2)? it is not commutative C=AXB -C=BXA A B 8

9. Sol. A vector is an object with three components that transform as follow under rotation in general, three independent angles are need to describe a rotation from one set to another. C (6,-4,4) A (3,-1,4) B (2,1,2) θ 9 D

10. X y z‾z‾ X‾X‾ θ θ π -θ X‾X‾ Y‾Y‾ Y‾Y‾ y x z θ π/2 -θ π -θ 10

11. A scalar (density, pressure, temperature, etc.) is a quantity whose specification (in any coordinate system) requires just one number. On the other hand, a vector (displacement, acceleration, force, etc.) Is a quantity whose specification requires three numbers, namely its components with respect to some basis. Scalars and vectors are both special cases of a more general called a tensor of order n whose specification in any coordinate system requires 3 n numbers, called the components of tensor. In fact, scalars are tensors of order zero with 3 0 =1 component. Vectors are tensors of order one with 3 1 = 3 components. tensor is an object with n component(nine component ) that transform as follow 11

12. n-Dimensional space: in three dimensional rectangular space, the coordinates of a point are(x,y,z). it convenient to write (x 1,x 2,x 3 ) for of a point in four dimensional space are given by (x 1,x 2,x 3,x 4 ).in general, the coordinates of a point in n-dimensional space are given by (x 1,x 2,x 3,x 4,…..,x n ) such n-dimensional space is denoted by V n. Superscript and subscript: In the symbol, the i,j written in the upper position are called superscripts and k,l written in the lower position are called subscripts. The Einstein's summation convention: Consider the sum of the. By using summation convention, drop the sigma sign and write convention as 12

13. This convention is called Einstein's summation convention and stated as "If a suffix ( اللاحقة )occurs twice in a term, once in the lower position and once in the upper position then that suffix implies sum over defined range." The range is not given, then assume that the range is form 1 to n. DUMMY INDEX: Any index which is repeated in given term is called a dummy index or dummy suffix. This is also called Umbral index. e.g Consider the expression a i x i where i is dummy index, then And These two equations prove that So any dummy index can be replaced by other index ranging the same numbers. 13

14. FREE INDEX: Any index occurring only once in a given term is called a FREE INDEX. e.g consider the expression where j is the free index. KRONECKER DELTA: The symbol called Kronecker Delta(a German mathematician Leopold Kronecker 1823 A.D) is defined by Similarly and are defined as and 14

15. Properties 1-if x 1,x 2,…,x n are independent coordinates, then, This implies that And also written as 2- (by summation convention) 3- (as j is dummy index) Preliminaries = In general, 4- 15

16. Exp.1 using summation convention. Solution Exp.2 (i) (ii) Solution (i) (as I and j are dummy indices) (ii) as m is dummy index. 16

17. Exp.3 If are constant and, calculate (i) (ii) solution (i) as as j is dummy index as given (ii)Differentiating it w.r.t x l : as 17

18. 18

19. Field and differential operator : A field is a physical entity that depends on one or more continuous parameters such a parameter can be viewed as a 'continuous index' that enumerates the infinitely many "coordinates" of the field. In particular, in a field that depends on the usual position vector x of R 3, each point in this space can be considered as one degree of freedom so that a field is a representation of a physical entity with an infinite number of degree of freedom.(By a field is meant a function of the position vector x.in physics, we used field to describe the state of continuous system). The gas has a density ρ, which is a function of position, the density is a scalar and is a scalar field. the gas at each point has some velocity which depends on position, the function v(x) is a vector field. 19

20. Given an arbitrary scalar field, we can construct the vector field The operator is called the gradient operator or del.op. it often written as Gradient :- the rate of change of the function in specified direction. we can use the gradient to find small changes in the function produced by small displacements. Consider two point separated by small displacement The difference between the value of the function at these two point is 20

21. Gradient of scalar :- is defined as the maximum rate of change of the scalar quantity and it’s direction is the direction of maximum change. therefore it is a vector quantity. Properties : 1- 2- 3- Ex. Find the value of when at the point (1,-2,-1)? ds Φ=constant Φ+dφ 21

22. Ex. Find the value of when at the point (1,-2,-1)? S0l. Integration of vector : There are three type of vector integral 1- Line integral : Let us suppose that we have a vector field as shown in the diagram. let A and B be any two point in the field. if we find out the value of from A to the point B along any path the quantity is known as line integral of from A to B. 22

23. If 1- 2- Ex: جد التكامل الخطي للمجال الممثل بالعلاقة عبر مسار معين الذي هو عبارة عن منحني على شكل قطع مكافئ تحدده العلاقة x=y 2 ما بين نقطة الأصل و النقطة (2p(2,√. Sol. θ A B 23

24. جد الشغل المبذول بتحريك جسم مرة واحدة حول الدائرة c الواقعة في المستوي xy) ) أذا كان مركزها عند نقطة الأصل ونصف قطرها 3 متر علما بأن قوة المجال هي. 24 Y=3 sin θ x=3 cos θ 3m

25. 2- Surface integral : The surface integral of a given surface S is equal to where da is the elementary area of unit vector n as shown in the diagram. Ex. Find the surface integral for the vector produced by surface 's' which part of circle lying in first quarter in xy - plane and its center in the origin and has radius 'a', 25

26. Sol. Since the surface is lying in the xy-plane Therefore, 26

27. جد التكامل السطحي للمجال مأخوذ فوق سطح مكعب طول ضلعه 1 سم وأحد إضلاعه يمر بنقطة الأصل. 27

28. 3- volume integral : We will have two volume integrals, one for scalar field and another for a vector field. if is scalar, the volume integral of over a volume V If is a vector, the volume integral of over a volume V EX: The interplanetary probe show in fig. is attached to the Sun by the force of magnitude F=-13x10 -22 /α 2 where α is the distance measured outward from sun to the probe. Determine how much work done by the Sum on the probe as the probe –Sun separation changes from 1.5*10 11 m to 2.3*10 11 m.. 28

29. Divergence of vector : The divergence of the vector is the limit of its surface integral per unit volume enclosed by the surface goes to zero, that is, In the rectangular coordinates the volume element. Provides a convenient basis for finding the explicit form of the divergence.If one corner of the rectangular parallelepiped is at the point,, then 29

30. 30 The limit is easily taken, and the divergence in rectangular coordinates is found to be

31. Properties : 1- 2- Divergence theorem (Gauss theorem ): The integral of divergence of vector over a volume V is equal to the surface integral of the normal component of the vector over the surface bounded V, This is Starting from the basic definition of the divergence, prove the divergence theorem. 31

32. 32 We can prove the theorem by first taking a volume in the shape of cubic (any other volume ) suppose that the cube occupies the region to, to, to then, the volume integral On other hand, if we perform the surface integral over the face of cube and over the face at, we obtain

33. Where the minus sign arise because, at, the direction of ds is opposite to that of. expression (2) agree with the first term in (1). the remaining term of (1) are exactly what we get from the surface integral over remaining face of the cube Curl of vector : The curl of a vector is the limit of the ratio of the integral of its cross product with the outward normal, over a closed surface, to the volume enclosed by the surface as the volume goes to zero, that is,. 33

34. 34 Properties : 1- 2- Stoke’s theorem : The line integral of a vector around a closed curve is equal to the integral of normal component of its curl over any surface bounded by the curve, that is,

35. H.W: starting from the basic definition of the curl, prove the Stoke's theorem Suppose an arbitrary open surface with boundary c. The unit vector is perpendicular to the surface. If the Fingers of your right hand lie a long dl, your thumb Will lie a long nˆ. We first carry out the proof for case of a flat surface in The shape of square consist of region x=x 0 to x 0 +l, y=y 0 to y 0 +l,(see fig b) But this is the same as over the four side of the square, hence H.W: prove that conservation field vector ? 35

36. Ex. Evaluate the integral where and ‘S’ is the surface of cubic bounded by the plane and the region through the corner of the cubic ? Sol. 36

37. EX. Verify the divergence theorem to the vector filed which was taken over a cubic where length 4cm and the origin passes through the center of the cubic and their side parallel to the axises (x,y,z). Sol. 37

38. Cylindrical and spherical coordinates :The relation between the Cylindrical and Cartesian coordinates and the Cylindrical coordinates and the unit vector (ρ,ф,z) as follows. 38

39. The relation between the spherical and Cartesian coordinates and the spherical coordinates and the unit vector as follows. 39