SECTION 1 Review of vectors Electricity 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
Vector algebra Adding/subtracting vectors:
A+B=C
A +B=C C
A A!B=D
C
A AD !B=D
A
B
!B
A
D
!B
B
Multiplication by a scalar:
A
A
2A 2A Unit vector:
BB ˆ = Bˆ B = B where B (which we sometimes write
B
as |B|) denotes the length of vector B
Dot product (or scalar product):
A !
B
A ! B =
AB cos!
The scalar product is the part of A which lies along B , times the magnitude of B (or equivalently the part of B which lies along A , times the magnitude of A).
The dot product is:
distributive
and
commutative,
meaning respectively that
A ! (B + C) = (A ! B) + (A ! C) A ! B = B ! A
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A!B = 0 A ! A = A2
If A is perpendicular to B, then Also, the dot product has the useful property that
Cross product (or vector product):
nˆ
B !
A
A ! B = (ABsin ! )nˆ
Geometrically, the cross product is the area of a parallelogram formed by 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.
A!A = 0
In particular:
Vector components It is often useful to write vectors in terms of their Cartesian (x, y, z), coordinates:
A = Ax xˆ + Ay yˆ + Az zˆ Here xˆ , yˆ , and zˆ 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 = (Ax + Bx )xˆ + (Ay + By ) yˆ + (Az + Bz ) zˆ Multiplication by a scalar: multiply each component by the scalar. aA = aAx xˆ + aAy yˆ + aAz zˆ
Products of vector in components Dot product: multiply the like components and sum.
! " B = A B + Ay By + Az Bz
x x Can be proved using results for unit vectors like:
xˆ ! xˆ = 1 xˆ ! yˆ = 0
Cross product: find the determinant of this matrix:
xˆ
yˆ
zˆ
A ! B = Ax
Ay
Az
Bx
By
Bz
or, equivalently,
A ! B = (A B " A B )xˆ + (Az Bx " Ax Bz ) yˆ + (Ax By " Ay Bx ) zˆ
y z y z Can be proved using results for unit vectors like:
ˆ
ˆ
x ! x = 0
xˆ ! yˆ = zˆ 2
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: n ˆ A ! (B " C) Geometrically, the triple product is the volume of the A parallelepiped formed by vectors A, B and C, but it has a sign C that can be positive or negative. B The sign of the triple product depends on the cyclic order of the vectors, so A ! (B " C) = B ! (C " A) = C ! (A " B) Those that are not in the same cyclic order have the opposite sign, e.g., B ! (C " A) = #B ! (A " C)
Position, displacement and separation vectors – notation (Griffiths) z
z
r
!
rˆ
r!"
(x, y, z)
source point !
r y x
r’ r re
re
!
field point
y
x re = r ! r" is called the source point: the place where the charge is located, is called the field point: the place where we want to know the electric field, 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 quantity (usually called “del”) as a vector, which is defined by
! = xˆ
" " " + yˆ + zˆ "x "y "z
This represents a vector because it has components defined in terms of the usual unit vectors along the x, y and z axes. However, instead of just numbers or scalar functions as its vector components, it has terms that differentiate. For example, means “partial differentiation with respect to x (so differentiate with respect to x ! while keeping the other variables y and z like constants) !x 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.
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Gradient of a scalar function In differential calculus with just one variable x, it is a simple property 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, y, z) in 3 dimensions the generalization is the gradient (written as ∇T or sometimes as grad T ) given by
!T = xˆ
"T "T "T + yˆ + zˆ "x "y "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 any point (x, y, z):
Ø The magnitude of the gradient ∇ T gives us the maximum rate of change of the function T at that point. Ø 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 xˆ + v yˆ + v zˆ
x y z and each component might depend on variables x, y and z in general, we get
!"v =
#vx #vy #vz + + #x #y #z
This scalar function, formed from vector v, is called the divergence of that vector function. It is denoted by ∇.v or sometimes div v.
Roughly 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 =
xˆ
yˆ
zˆ
# #x
# #y
# #z
vx
vy
vz
This new vector is called the curl of the function v. It is denoted by ∇ × v or sometimes curl v.
The determinant can be multiplied out in the usual way, so for example the x-component of the curl is
# !v !v & xˆ % z " y ( $ !y !z '
Roughly the curl calculates whether the “flow” of a function at any position is rotational or not.
Product rules in vector calculus These are quoted for reference only (so do not bother to memorize them!). They 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
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For gradient:
For divergence:
!( fg) = f !g + g!f !(A " B) = A # (! # B) + B # (! # A) + (A " !)B + (B" !)A ! " ( fA) = f (! " A) + A " (!f )
!(A " B) = B # (! " A) $ A # (! " B)
For curl: ! " ( fA) = f (! " A) # A " (!f )
! " (A " B) = (B # !)A $ (A # !)B + A(! # B) $ B " (! # A) Here scalar functions f and g and the components of vector functions A, B and C can depend on x, y, and z.
Second derivatives involving ∇ These can occur in several different ways. We know that gradient ∇T is a vector, so we could find its divergence or curl.
Taking the divergence gives
#2T #2T #2T ! " (!T ) = 2 + 2 + 2 #x #y #z 2
The right-hand side is often written as ∇ T , where we define the operator ∇2 which is called the Laplacian by
!2 =
"2 "2 "2 + + "x 2 "y 2 "z 2
Next, if we try taking the curl of gradient ∇T, we always get 0. It means that for any 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 very often in physics, so we will not work it out. Although it looks similar to the Laplacian, they are not the same: 2
!(! " v) # ! v The divergence of the curl, like the curl of a gradient, is always 0:
! " (! # v) = 0
Finally, the curl of the curl can be rewritten in terms of other quantities already discussed using:
! " (! " v) = !(! # v) $ ! 2 v
Integral vector calculus
Basic integrals involve just one variable (usually x) and are xn =b b taken between limits along the coordinate axis. lim f (xn )! xn = f (x) dx a ! xn !0 Recall that integrals can always be regarded as the limiting xn =a cases of sums: 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 they are like 2-dimensional analogs)
"
#
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•
Volume integrals –
these are taken over a volume rather than a line (so they are like 3-dimensional analogs)
Line integrals
In general, a line integral can be along any three-dimensional line. In this course, they will usually be in straight line or circular segments. A typical form involves a dot product like
End b
Here vector v might be a function of coordinates x, y and z, and the vector element of length dl is the vector with components dx, dy, and dz. So v.dl = vxdx + vydy + vzdz
Element of length
Start a
Sometimes the integral path forms a closed loop, in which case it’s written as: " v ! dl ! A familiar line integral from first year physics is the work W done by a force:
W=
"
F ! dl
Surface integrals
A surface integral over a general surface in 3-dimensions is typically of the form:
Normal vector element da
The vector element of area, da, will have components like Element of area on the surface
(because the x direction is normal to the yz 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 physics typically involve vector functions (in a dot product). Volume integrals more commonly occur with scalar functions: where T is a scalar function of x, y and z, and dτ is a volume element:
Volume integrals can also be done on vector functions: each cartesian component would then be integrated separately.
Fundamental theorem of calculus A fundamental theorem of calculus states that for a scalar function f(x):
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You can think of this as essentially saying that if you start with a function f(x) and differentiate it, then if you integrate again you get the function you 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 very direct generalization of the previous fundamental theorem:
#
b
(!T )" dl = T (a) $ T (b)
a [ path] Here T(x,y,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 any path provided the end points a and b are the same.
An important corollary is that the integral of a gradient around a closed loop is always 0 (because T(a) and T(b) will cancel out when a = b):
!# (!T )" dl = 0
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):
#
(! " v) d! =
!#
v " da
[surface] [volume] In other words, the total divergence of a vector function integrated throughout a particular volume can be found by adding up (integrating) the net “flow” in or out through the closed surface bounding that volume.
Fundamental theorem for curl
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 boundary line of that area:
Boundary path
Two corollaries are that:
!#
#
[surface]
[surface]
(! " v) $ da
(! " v) $ da =
It is also referred to as Stokes Theorem.
depends only on the boundary line, not the surface chosen. for any closed surface. 0
Curvilinear coordinates Spherical coordinates
z
Angle "
In basic electrostatics problems, we often have a point charge or a spherically symmetric charge distribution. In these cases, it makes sense to use spherical coordinates (r, θ, ϕ) for vector r rather than Cartesian coordinates (x, y, z).
r y x
Angle !
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We also need to introduce new unit vectors instead of unit vectors in the Cartesian system:
z
Angle "
We define three new unit vectors:
!ˆ
rˆ
rˆ
r
y
!ˆ x
Angle !
The unit vectors are related by Unit vector
We will sometimes need results for infinitesimal displacements in the position. (These were simply dx, dy, and dz in the Cartesian system).
rˆ
- This in the direction away from the origin
!ˆ
- This is in the direction of increasing !
!ˆ
- This is in the direction of increasing !
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.
dr
Displacement
dr rd!
!ˆ
rd!
d! r sin ! d"
!ˆ
r sin ! d"
!
d!
r sin !
Vector Derivatives in Spherical Coordinates It follows that the earlier definition of the del operator as becomes replaced in spherical coordinates by The expressions for gradient, divergence and curl can be worked out from the above (see Appendix 1A in the
notes, but do not memorize).
Cylindrical coordinates
z
zˆ s z
In some cases, e.g., when we have a line charge or a cylindrically symmetric distribution, it makes sense to use cylindrical polar coordinates (s, ϕ, z).
!ˆ sˆ y
x
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The three unit vectors are shown:
(in direction of increasing s) sˆ = cos ! xˆ + sin ! yˆ
!ˆ = !sin ! xˆ + cos ! yˆ zˆ = zˆ
(in direction of increasing !) (in direction of increasing z)
Unit vector
Again, we will sometimes need results for infinitesimal displacements in the position. (These were simply dx, dy, and dz in the Cartesian system).
Displacement
sˆ
ds
!ˆ
sd!
zˆ
dz
Vector Derivatives in Cylindrical 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).
Dirac delta function
Example – calculation of a divergence Consider the vector function: It points radially outwards from the origin, and obviously it diverges at 0. Suppose we calculate its divergence:
!"v =
1 #$ 2 1' 1 # &r )= (1) = 0 r 2 #r % r 2 ( r 2 #r
Suppose now we do it another way by applying the divergence theorem, and integrating the divergence over a sphere of any arbitrary radius R:
# 1 & rˆ ! R 2 sin ! d! d" rˆ ) = 2 ( ( '
!" v ! da = " %$ R
("
# 0
sin ! d!
)( "
2# 0
)
d " = 4#
for any 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 infinitely high spike that is located at x = 0 but is equal to zero everywhere else. Its other defining property is that the area under the spike is unity. and 9
If we multiply any continuous function f(x) by a delta function, the product is 0 everywhere except at x = 0, and so f (x)! (x) = f (0)! (x) If we integrate over any range that includes 0, we get
#
"
f (x)! (x)dx = f (0) #
"
! (x)dx = f (0)
!" !" We can easily generalize the delta function by shifting the spike’s position:
Also
$& " ! (x ! a) = % &' 0
if x = a if x # a
" !"
! (x ! a)dx = 1
#
f (x)! (x ! a) = f (a)! (x ! a)
#
and
" !"
f (x)! (x ! a)dx = f (a)
Three-‐dimensional Dirac delta function The generalization to 3 dimensions is straightforward:
! 3 (r) = ! (x)! (y)! (z)
Now it is the 3-dimensional integral that is unity:
!
! 3 (r)d" = allspace
"
allspace
Also, for a 3-dimensional function f (r):
#
#
#
"#
"#
"#
! ! !
! (x)! (y)! (z) dxdydz = 1
f (r)! 3 (r ! a)d" = f (a)
Now we can revisit the divergence paradox concerning
We already found that the divergence is 0 everywhere 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):
z
source point !
r!"
re
r
!
We can also show that
field point
y
x
re = r ! r"
and therefore
Vector fields
Later we will define an electric field vector E and a magnetic field vector B. Eventually, 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
1st 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 ! " F = #! " !V = 0 Less obviously, it can also be proved the other way round: ! " F = 0 # F = $!V
2nd case: Similarly, 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
F = ! " A
! " F = ! " (! # A) = 0 Again, it can also be proved the other way round:
! " F = 0 # F = ! $ A Solid angles
Solid angles are useful when we need a measure of all the spatial directions subtended at the vertex of any cone, even if it has an irregular shape. Area A on surface of sphere
R
Solid angle Ω
The definition is as follows: Draw a sphere of any radius R centred at the vertex and find the area A intersected on the curved surface of the sphere. The solid angle Ω is
Since A is proportional to R2, 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 (by analogy 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).
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APPENDIX 1A (for reference)
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