Vectors 向量 Chapter 16 Gialih Lin, Ph. D. Professor

16.1 Concepts Magnitude only called scalar quantities Vectors require both magnitude and direction Null vector 0, a=0

16.2 Vector algebra Equality a=b Vector addition Commutative rule 交換律 a+b=b+a Parallelogram law 平行四 邊形律 Subtraction 減法 a+(-b)=a-b Scalar multiplication a+a=2a Unit vector, â= a/ ｜ a ｜

16.3 Components of vectors A (x 1,y 1 ), B(x 2,y 2 ) AB = a = (x 2 -x 1,y 2 -y 1 ) = (x 2 -x 1 ) i+ (y 2 -y 1 ) j = (x 2 -x 1 ) (1,0)+ (y 2 -y 1 ) (0,1) The length of the vector, ｜ a ｜ = [(x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 ] 1/2 Equality Addition Scalar multiplication

Ordinary geometry Vector, r r = a 1 i+a 2 j+a 3 k Column vector

Example 16.5 The center of mass of a system of N masses, m 1 with position vector r 1, m 2 at r 2 R= (m 1 r 1 +m 2 r 2 +…+m N r N )/M Where M=m 1 +m 2 +…+m N is the total mass R(X,Y,Z) X= (m 1 x 1 +m 2 x 2 +…+m N x N )/M Y= (m 1 y 1 +m 2 y 2 +…+m N y N )/M Z= (m 1 z 1 +m 2 z 2 +…+m N z N )/M

Example 16.6 Dipole moments 偶極 距 The system of two charges with =q at r 1 and q at r 2, defines an electric dipole with vector dipole moment  =-qr 1 +qr 2 =q(r 2 -r 1 ) =qr In general  =q 1 r 1 +q 2 r 2 +…+q N r N =  1 +  2 +…+  N The total dipole moment of a molecular is sometimes interpreted as the sum of bond moments.

Base vector i= (1,0,0) j= (0,1,0) k= (0,0,1) a = a x i+a y j+a z k = a x (1,0,0) +a y (0,1,0) +a z (0,0,1) = (a x,0,0) + (0,a y,0) + (0,0,a z ) = (a x,a y,a z )

16.5 The scalar (dot) product 內積 a ． b=ab cos  Projection of a on b a ． b = 0 The vectors are said to be orthogonal 垂直 to each other.  =  /2 a ． a= a x 2 +a y 2 +a z 2 = ｜ a ｜ 2 ｜ a ｜ = ( a ． a ) 1/2

The use of cartesian base vector Unit vector I, j, and k a= (a x,a y,a z ) b= (b x,b y,b z ) a ． b = a x b x + a y b y + a z b z

Example Force and work W=F ． D=Fd cos  W=∫F (r) ． dr

16.6 The vector (cross) product v=a x b ∣ v ∣ =a b sin  Vector product is not commultative; it is anti- commutative bxa = -axb

11-2 Vector Cross Product; Torque as a Vector The vector cross product is defined as: The direction of the cross product is defined by a right-hand rule:

11-2 Vector Cross Product; Torque as a Vector The cross product can also be written in determinant form:

11-2 Vector Cross Product; Torque as a Vector Some properties of the cross product:

Example 16.6 Moment of force (torque) T=rxF

11-2 Vector Cross Product; Torque as a Vector Torque can be defined as the vector product of the force and the vector from the point of action of the force to the axis of rotation:

Example An electric dipole in an electric field An electric dipole  =qr (see example 16.6) in an electric field E T =  xE

16.7 Scalar and vector fields A function of the coordinates ofa point in space is called a function of position or field. A scalar function of position, a scalar field, f = f (r)= f(x,y,z)

Atomic units Schrödinger Equation for the motion of the stationary nucleus in the hydrogen atom m the rest mass of the electron e the charge on the proton h Plank’s constant ħ = h/ 2   o the permittivity of a vacuum ▽ : gradient see p467 grad f = ▽ = (  f/  x)I + (  f/  y) j + (  f/  z) k Unit of x-axis I = (1,0,0); unit of y-axis j=(0,1,0);unit of z-axis k=(0,0,1) The Laplacian operator ▽ 2 = (  2 /  x 2 ) + (  2 /  y 2 ) + (  2 /  z 2 )

16.8 The gradient of a scalar field ▽ : gradient see p467 grad f = ▽ = (  f/  x)I + (  f/  y) j + (  f/  z) k Unit of x-axis I = (1,0,0); unit of y-axis j=(0,1,0);unit of z-axis k=(0,0,1) The Laplacian operator see p270 ▽ 2 = (  2 /  x 2 ) + (  2 /  y 2 ) + (  2 /  z 2 )

Example Coulomb forces

Experiment shows that the electric force between two charges is proportional to the product of the charges and inversely proportional to the distance between them Coulomb’s Law

Coulomb’s law: This equation gives the magnitude of the force between two charges Coulomb’s Law

The force is along the line connecting the charges, and is attractive if the charges are opposite, and repulsive if they are the same Coulomb’s Law

Unit of charge: coulomb, C. The proportionality constant in Coulomb’s law is then: k = 8.99 x 10 9 N·m 2 /C 2. Charges produced by rubbing are typically around a microcoulomb: 1 μC = C Coulomb’s Law

Charge on the electron: e = x C. Electric charge is quantized in units of the electron charge Coulomb’s Law

The proportionality constant k can also be written in terms of ε 0, the permittivity of free space: 21-5 Coulomb’s Law

Conceptual Example 21-1: Which charge exerts the greater force? Two positive point charges, Q 1 = 50 μC and Q 2 = 1 μC, are separated by a distance. Which is larger in magnitude, the force that Q 1 exerts on Q 2 or the force that Q 2 exerts on Q 1 ?

21-5 Coulomb’s Law Example 21-2: Three charges in a line. Three charged particles are arranged in a line, as shown. Calculate the net electrostatic force on particle 3 (the -4.0 μC on the right) due to the other two charges.

21-5 Coulomb’s Law Example 21-3: Electric force using vector components. Calculate the net electrostatic force on charge Q 3 shown in the figure due to the charges Q 1 and Q 2.

21-5 Coulomb’s Law Conceptual Example 21-4: Make the force on Q 3 zero. In the figure, where could you place a fourth charge, Q 4 = -50 μC, so that the net force on Q 3 would be zero?

16.9 Divergence and cur of a vector field div v= ▽ ． v = (  f/  x) + (  f/  y) + (  f/  z) v= ▽ f div v= ▽ ． ▽ f = [(  /  x)i + (  /  y)j + (  /  z) k] ． [(  f/  x)i + (  f/  y)j + (  f/  z) k]= (  2 f /  x 2 ) + (  2 f /  y 2 ) + (  2 f /  z 2 ) div v=The Laplacian operator see p270 ▽ 2 = (  2 /  x 2 ) + (  2 /  y 2 ) + (  2 /  z 2 )

16.10 Vector spaces The vectors discussed in this chapter are three-dimensional vectors, or vectors in a three-dimensional vector space. The concept of vector can be extended to any number of dimensions by defining vectors in n dimensions as quantities that have n components and that obey the laws of vector algebra described in Sections 16.2 and 16.3.

Quantum mechanics Any function, , can be expressed in terms of its components for a set of basis functions,  i, of unit length  = a 1  1 +a 2  2 +a 3  3 + ….+a n  n  = b 1  1 +b 2  2 +b 3  3 + ….+b n  n

Scalar product Projection of vector, r, onto vector, s, is called scalar product of r and s r ． s = a 1 b 1 +a 2 b 2 +a 3 b 3

Scalar product or inner product of  and , = a 1 b 1 +a 2 b a n b n