SECTION 1 Review of vectors

Transcription

1 SECTION 1 Review of vectors Electricit and magnetism involve fields in 3D space. This section, based on Chapter 1 of Griffiths, reviews the tools we need to work with them. The topics are: Vector algebra Differential calculus Integral calculus Curvilinear coordinates Dirac delta function Vector fields Solid angles Adding/subtracting vectors: Vector algebra A B = D A A + B = C C D B A B Multiplication b a scalar: A 2A Unit vector: B B = ˆB where B (which we sometimes write as B ) denotes the length of vector B Dot product (or scalar product): A B A B = ABcos The scalar product is the part of A which lies along B, times the magnitude of B (or equivalentl the part of B which lies along A, times the magnitude of A). The dot product is: distributive and commutative, meaning respectivel that A (B + C) = (A B)+ (A C) A B = B A 1

2 If A is perpendicular to B, then Also, the dot product has the useful propert that Cross product (or vector product): ˆn B A B = 0 A A = A 2 A A B = (ABsin) ˆn Geometricall, the cross product is the area of a parallelogram formed b A and B. It is in the direction of the unit vector perpendicular to both A and B, with the right hand rule determining the sign. The cross product is: distributive but not commutative: A (B + C) = (A B)+ (A C) A B = "(B A) If A is parallel to B, the cross product is zero. In particular: Vector components It is often useful to write vectors in terms of their Cartesian (x,, z), coordinates: A = A x ˆx + A ŷ + A z ẑ Here ˆx, ŷ, and ẑ are defined as unit vectors in the directions of the axes. Now we can reformulate the algebra rules in terms of components: Addition: add the like components. A + B = (A x + B x ) ˆx + (A + B )ŷ + (A z + B z )ẑ Multiplication b a scalar: multipl each component b the scalar. aa = aa x ˆx + aa ŷ + aa z ẑ Products of vector in components Dot product: multipl the like components and sum. " B = A x B x + A B + A z B z Can be proved using results for unit vectors like: ˆx ˆx =1 ˆx ŷ = 0 Cross product: find the determinant of this matrix: A A = 0 A B = ˆx ŷ ẑ A x A A z B x B B z or, equivalentl, A B = (A B z " A B z ) ˆx + (A z B x " A x B z )ŷ + (A x B " A B x )ẑ Can be proved using results for unit vectors like: ˆx ˆx = 0 ˆx ŷ = ẑ 2

3 Scalar triple products of vectors Since the cross product of two vectors is another vector, we can take the dot product of this with a third vector: A (B " C) ˆn A C B The sign of the triple product depends on the cclic order of the vectors, so A (B " C) = B(C " A) = C(A " B) Those that are not in the same cclic order have the opposite sign, e.g., B(C " A) = #B(A " C) Geometricall, the triple product is the volume of the parallelepiped formed b vectors A, B and C, but it has a sign that can be positive or negative. Position, displacement and separation vectors notation (Griffiths) z x r ˆr (x,, z) x z r" source point r r e = r r" r e field point r is called the source point: the place where the charge is located, r is called the field point: the place where we want to know the electric field, re is called the separation vector: it specifies the distance and direction from the source to the point where we want to calculate the field. Differential vector calculus The operator We can treat this quantit (usuall called del ) as a vector, which is defined b = ˆx " "x + ŷ " " + ẑ " "z This represents a vector because it has components defined in terms of the usual unit vectors along the x, and z axes. However, instead of just numbers or scalar functions as its vector components, it has terms that differentiate. For example, x means partial differentiation with respect to x (so differentiate with respect to x while keeping the other variables and z like constants) So if there is a scalar function to the right side of the del operator, it will give a vector and each component will be a partial derivative of that function. Also, because del is an operator, we can do other operations with it, like taking the cross product or the dot product of a vector function. 3

4 Gradient of a scalar function In differential calculus with just one variable x, it is a simple propert that the derivative df/dx of a function f(x) measures the slope or gradient of the curve when function f(x) is plotted versus x. For a scalar function T(x,, z) in 3 dimensions the generalization is the gradient (written as T or sometimes as grad T ) given b T = ˆx "T "x + ŷ "T " + ẑ "T "z Note that the gradient is obtained when operates on a scalar function T in 3 dimensions, but the result of doing this is a vector, calculated for an point (x,, z): Ø Ø The magnitude of the gradient T gives us the maximum rate of change of the function T at that point. The direction of gradient T gives us the direction of maximum rate of change. Divergence of a vector If we take the dot product.v of the del operator with a vector function v, where v = v x ˆx + v ŷ + v z ẑ and each component might depend on variables x, and z in general, we get " v = #v x #x + #v # + #v z #z This scalar function, formed from vector v, is called the divergence of that vector function. It is denoted b.v or sometimes div v. Roughl it measures whether there are sources or sinks of the function at the point where we calculate the divergence. Curl of a vector If we now take the cross product of with a vector function v, we have: " v = ˆx ŷ ẑ # #x # # # #z v x v v z This new vector is called the curl of the function v. It is denoted b v or sometimes curl v. The determinant can be multiplied out in the usual wa, so for example the x-component of the curl is # ˆx v z " v & % ( $ z ' Roughl the curl calculates whether the flow of a function at an position is rotational or not. Product rules in vector calculus These are quoted for reference onl (so do not bother to memorize them). The will be provided for all midterms and exams. Sometimes we need to calculate grad, div or curl of products of two functions (either scalar or vector). Some useful results are 4

5 For gradient: For divergence: For curl: Here scalar functions f and g and the components of vector functions A, B and C can depend on x,, and z. Second derivatives involving These can occur in several different was. We know that gradient T is a vector, so we could find its divergence or curl. Taking the divergence gives ( fg) = f g + gf (A " B) = A # ( # B) + B # ( # A) + (A " )B + (B" )A "( fa) = f ( " A)+ A "(f ) (A " B) = B#( " A) $ A #( " B) " ( fa) = f ( " A)# A " (f ) " (A " B) = (B# )A $ (A # )B + A( # B) $ B " ( # A) "(T ) = #2 T #x + #2 T 2 # + #2 T 2 #z 2 The right-hand side is often written as 2 T, where we define the operator 2 which is called the Laplacian b 2 = "2 "x 2 + "2 " 2 + "2 "z 2 Next, if we tr taking the curl of gradient T, we alwas get 0. It means that for an scalar function T we have " (T ) = 0 Another possible second derivative is the gradient of the divergence of a vector v. This is ( " v) It does not occur ver often in phsics, so we will not work it out. Although it looks similar to the Laplacian, the are not the same: ( " v) # 2 v The divergence of the curl, like the curl of a gradient, is alwas 0: "( # v) = 0 Finall, the curl of the curl can be rewritten in terms of other quantities alread discussed using: " ( " v) = ( # v)$ 2 v Integral vector calculus Basic integrals involve just one variable (usuall x) and are taken between limits along the coordinate axis. Recall that integrals can alwas be regarded as the limiting cases of sums: lim x n 0 b " f (x n )x n = f (x)dx We will be using three kinds of integrals involving vectors: Line integrals these are along a line, as in the above example, but it doesn t have to be a straight line in general Surface integrals these are taken over an area rather than a line (so the are like 2-dimensional analogs) x n =b x n =a # a 5

6 Volume integrals these are taken over a volume rather than a line (so the are like 3-dimensional analogs) Line integrals End b In general, a line integral can be along an three-dimensional line. In this course, the will usuall be in straight line or circular segments. A tpical form involves a dot product like Start a Element of length Here vector v might be a function of coordinates x, and z, and the vector element of length dl is the vector with components dx, d, and dz. So v.dl = v x dx + v d + v z dz Sometimes the integral path forms a closed loop, in which case it s written as: v d " l A familiar line integral from first ear phsics is the work W done b a force: W = " F dl Surface integrals Normal vector element da A surface integral over a general surface in 3-dimensions is tpicall of the form: Element of area on the surface The vector element of area, da, will have components like (because the x direction is normal to the z plane). As with line integrals, we often do closed integrals: in this case the surface encloses a volume (e.g., the surface area around a sphere) " v da Volume integrals Volume of integration Volume element d Line integrals and surface integrals in phsics tpicall involve vector functions (in a dot product). Volume integrals more commonl occur with scalar functions: where T is a scalar function of x, and z, and dτ is a volume element: Volume integrals can also be done on vector functions: each cartesian component would then be integrated separatel. Fundamental theorem of calculus A fundamental theorem of calculus states that for a scalar function f(x): 6

7 You can think of this as essentiall saing that if ou start with a function f(x) and differentiate it, then if ou integrate again ou get the function ou started with (evaluated between the integration limits). There are three analogous fundamental theorems for vector calculus: one for gradient, one for divergence, and one for curl. Fundamental theorem for gradient For gradients, there is a ver direct generalization of the previous fundamental theorem: Here T(x,,z) is a scalar function of position and the integral is a line integral along a path from point a to point b. Notice that the right hand side doesn t seem to depend on the path chosen; the result of the line integral is the same for an path provided the end points a and b are the same. An important corollar is that the integral of a gradient around a closed loop is alwas 0 (because T(a) and T(b) will cancel out when a = b): Fundamental theorem for divergence For divergences, the result relates a volume integral to a surface integral (taken over the total surface area of that volume): In other words, the total divergence of a vector function integrated throughout a particular volume can be found b adding up (integrating) the net flow in or out through the closed surface bounding that volume. Fundamental theorem for curl b # (T )" dl = T(a)$T(b) a [ path] # (T )" dl = 0 # ( " v) d = v " da [volume] # [surface] Boundar path Area For curls, the result relates the curl of a vector, when integrated over a specified area, to the line integral of the vector taken around the boundar line of that area: It is also referred to as Stokes Theorem. Two corollaries are that: # ( " v) $ da [surface] # ( " v) $ da = 0 [surface] Spherical coordinates depends onl on the boundar line, not the surface chosen. for an closed surface. Curvilinear coordinates z Angle " r In basic electrostatics problems, we often have a point charge or a sphericall smmetric charge distribution. In these cases, it makes sense to use spherical coordinates (r, θ, ϕ) for vector r rather than Cartesian coordinates (x,, z). x Angle 7

8 We also need to introduce new unit vectors instead of unit vectors in the Cartesian sstem: z Angle " We define three new unit vectors: ˆ r ˆr ˆr - This in the direction awa from the origin ˆ ˆ - This is in the direction of increasing x Angle ˆ - This is in the direction of increasing The unit vectors are related b Note that, unlike the Cartesian unit vectors, the directions of the spherical unit vectors depend on the direction of vector r relative to the origin. Unit vector Displacement dr We will sometimes need results for infinitesimal displacements in the position. (These were simpl dx, d, and dz in the Cartesian sstem). ˆr ˆ dr rd d rd rsind" ˆ rsind" d Vector Derivatives in Spherical Coordinates It follows that the earlier definition of the del operator as rsin becomes replaced in spherical coordinates b The expressions for gradient, divergence and curl can be worked out from the above (see Appendix 1A in the notes, but do not memorize). Clindrical coordinates z ẑ ˆ s z ŝ In some cases, e.g., when we have a line charge or a clindricall smmetric distribution, it makes sense to use clindrical polar coordinates (s, ϕ, z). x 8

9 The three unit vectors are shown: ŝ = cos ˆx + sin ŷ ˆ = sin ˆx + cos ŷ ẑ = ẑ (in direction of increasing s) (in direction of increasing ) (in direction of increasing z) Again, we will sometimes need results for infinitesimal displacements in the position. (These were simpl dx, d, and dz in the Cartesian sstem). Unit vector ŝ ˆ ẑ Displacement ds sd dz Vector Derivatives in Clindrical Coordinates It follows that the expression for the del operator becomes The expressions for gradient, divergence and curl can be worked out from the above (see Appendix 1A in the notes, but do not memorize). Example calculation of a divergence Consider the vector function: Dirac delta function It points radiall outwards from the origin, and obviousl it diverges at 0. Suppose we calculate its divergence: " v = 1 # $ 1 ' & r 2 #r r2 ) = 1 # ( % r 2 ( r 2 #r 1 ) = 0 Suppose now we do it another wa b appling the divergence theorem, and integrating the divergence over a sphere of an arbitrar radius R: # 1 " v da = R ˆr & " % ( R 2 # 2# ( sindd" ˆr ) = $ 2 ' ( " sin d 0 )( " d" 0 ) = 4# for an R There seems to be a problem, because one result gives zero divergence while the other gives nonzero In particular, what s happening to the divergence at the origin? The Dirac delta function The one-dimensional Dirac delta function represents an infinitel high spike that is located at x = 0 but is equal to zero everwhere else. Its other defining propert is that the area under the spike is unit. and 9

10 If we multipl an continuous function f(x) b a delta function, the product is 0 everwhere except at x = 0, and so f (x)(x) = f (0)(x) If we integrate over an range that includes 0, we get " " # f (x)(x)dx = f (0) # (x)dx = f (0) " " We can easil generalize the delta function b shifting the spike s position: Also $ & (x a) = % '& " if x = a 0 if x # a Three- dimensional Dirac delta function The generalization to 3 dimensions is straightforward: Now it is the 3-dimensional integral that is unit: 3 # # # (r) d" = (x)()(z) dxddz =1 allspace "# "# "# Also, for a 3-dimensional function f (r): " f (r) 3 (r a) d" = f (a) Now we can revisit the divergence paradox concerning and (x a)dx =1 " f (x)(x a) = f (a)(x a) # f (x)(x a)dx = f (a) " 3 (r) = (x)()(z) allspace We alread found that the divergence is 0 everwhere except the origin, and the corresponding surface integral is 4π. It leads to the conclusion that: # " " In general, in terms of the displacement vector (defined earlier as the distance from the source point to the field point): x z r" and therefore source point r r e = r r" r e field point We can also show that Vector fields Later we will define an electric field vector E and a magnetic field vector B. Eventuall, we will arrive at Maxwell s equations, which tell us about the divergence and curl of E and B. Their properties make it useful to define some potentials (analogous to what is often done in classical mechanics). 10

11 1 st case: If a vector field F can be written as the gradient of a function V (called the scalar potential), meaning F = "V then it is obvious that Less obviousl, it can also be proved the other wa round: 2 nd case: Similarl, if a vector field F can be written as the curl of a function A (called the vector potential), so that then it is obvious that Again, it can also be proved the other wa round: Solid angles Solid angles are useful when we need a measure of all the spatial directions subtended at the vertex of an cone, even if it has an irregular shape. Area A on surface of sphere " F = # " V = 0 " F = 0 # F = $V F = " A " F = "( # A) = 0 " F = 0 # F = $ A The definition is as follows: Draw a sphere of an radius R centred at the vertex and find the area A intersected on the curved surface of the sphere. The solid angle Ω is R Solid angle Ω Since A is proportional to R 2, the result for Ω does not depend on the choice for R. It follows that the total solid angle for all directions in space is The definition of solid angle makes Ω dimensionless, but the unit of steradian is sometimes used (b analog with radian for angles). This definition will be useful later in proving Gauss s law for the electric field. Example: Consider a regular solid cone that has half-angle θ at the vertex. Prove that the total solid angle subtended at the vertex is = 2 (1" cos") (To be done as an example in classes). 11

12 APPENDIX 1A (for reference) 12