Introduction to Electrodynamics - HEP

Introduction to Electrodynamics
R. D. Reece
November 2, 2005
1 Vector Calculus
1.1 Definitions
Vector Addition:
Multiplication by a scalar:
Dot or Scalar Product:
⃗A + ⃗ B ≡ (A x + B x )ˆx + (A y + B y )ŷ + (A z + B z )ẑ (1)
φ ⃗ A ≡ φA xˆx + φA y ŷ + φA z ẑ (2)
⃗A · ⃗B = A x B x + A y B y + A z B z (3)
= ∣A
⃗ ∣ ∣B
⃗ ∣ cos θ (4)
Cross or Vector Product:
⃗A × ⃗ B =
=
∣ ˆx ŷ ẑ ∣∣∣∣∣ A x A y A z
(5)
∣ B x B y B z ∣A
⃗ ∣ ∣B
⃗ ∣ ⃗n sin θ (6)
1.2 Commuting
⃗A · ⃗B = ⃗ B · ⃗A (7)
⃗A × ⃗ B = − ⃗ B × ⃗ A (8)
1.3 Triple Products
⃗A · ( ⃗ B × ⃗ C) = ⃗ B · ( ⃗ C × ⃗ A) = ⃗ C · ( ⃗ A × ⃗ B) (9)
⃗A × ( ⃗ B × ⃗ C) = ⃗ B( ⃗ A · ⃗C) − ⃗ C( ⃗ A · ⃗B) (10)
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1.4 Vector Derivatives in Cartesian Coordinates
Gradient:
Divergence:
Curl:
Laplacian of a scalar:
Laplacian of a vector:
1.5 Product Rules
∇φ = ∂φ ∂φ ˆx +
∂x ∂y ŷ + ∂φ
∂z ẑ (11)
∇ · ⃗v = ∂⃗v
∂x + ∂⃗v
∂y + ∂⃗v
∂z
(12)
∣ ˆx ŷ ẑ ∣∣∣∣∣
∇ × ⃗v =
∂ ∂ ∂
∂x ∂y ∂z
(13)
∣ v x v y v z
∇ 2 φ ≡ ∇ · ∇φ (14)
= ∂2 φ
∂x 2 + ∂2 φ
∂y 2 + ∂2 φ
∂z 2 (15)
∇ 2 ⃗v = ˆx∇ 2 v x + ŷ∇ 2 v y + ẑ∇ 2 v z (16)
∇(fg) = f(∇g) + g(∇f) (17)
∇( ⃗ A · ⃗B) = ⃗ A × (∇ × ⃗ B) + ⃗ B × (∇ × ⃗ A) + ( ⃗ A · ∇) ⃗ B + ( ⃗ B · ∇) ⃗ A(18)
∇ · (f ⃗ A) = f(∇ · ⃗A) + ⃗ A · (∇f) (19)
∇ · ( ⃗ A × ⃗ B) = ⃗ B · (∇ × ⃗ A) − ⃗ A · (∇ × ⃗ B) (20)
∇ × (f ⃗ A) = (21)
∇ × ( ⃗ A × ⃗ B) = (22)
1.6 Second Derivatives
1.7 Vector Implications
1.8 Taylor’s Theorem
For f : R → R:
f(x 0 + x) =
∞∑
k=0
f (k) (x 0 )
x k =
k!
k=0
∞∑
k=0
1
k!
d k f(x)
dx k ∣
∣∣∣x=x0
x k (23)
For f : R n → R, in vector notation:
∞∑ 1
f(⃗r 0 + ⃗r) =
k! (⃗r · ∇| ⃗r= ⃗r 0
) k f(⃗r) (24)
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1.9 Fundamental Theorems
The Fundamental Theorem of Calculus:
∫ b
The Fundamental Theorem of Gradients:
∫ ⃗a
a
df
dx = f(b) − f(a) (25)
dx
⃗ b
∇f(⃗r) · d ⃗ l = f( ⃗ b) − f(⃗a) (26)
Stokes’ Theorem or The Curl Theorem:
∫
∮
∇ × f(⃗r) ⃗ · dS ⃗ =
⃗f(⃗r) · d ⃗ l (27)
Gauss’ Theorem or The Divergence Theorem:
∫
∮
∇ · ⃗f(⃗r) dV = ⃗f(⃗r) · dS ⃗ (28)
Generalized Stokes’ Theorem:
∫ ∫
df = f (29)
m ∂m
1.10 The Laplacian
The Laplacian operator acting on a scalar φ is denoted and defined as:
∇ 2 φ ≡ ∇ · ∇φ (30)
which reduces to following in R n :
∇ 2 φ =
n∑
k=0
∂ 2 φ
∂x 2 k
(31)
in R 3 :
∇ 2 φ = ∂2 φ
∂x 2 + ∂2 φ
∂y 2 + ∂2 φ
∂z 2 (32)
To demonstrate demonstrate what the Laplacian means, consider the average
value of φ in a cube centered at the origin and with sides of length a,
〈φ〉 = 1 V
Taylor expanding φ gives:
[
φ = φ 0 + x ∂φ
∂x + y ∂φ
∂y + z ∂φ ]
∂z
∫
φ dV = 1 ∫∫∫ a/2
a 3 φ dx dy dz (33)
−a/2
φ=φ 0
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+ 1 [
]
x 2 ∂2 φ
2 ∂x 2 + y2 ∂2 φ
∂y 2 + z2 ∂2 φ
∂z 2 + 2xy ∂2 φ
∂x∂y + 2xz ∂2 φ
∂x∂z + 2yz ∂2 φ
+ ...
∂y∂z
φ=φ 0
(34)
After plugging this expansion into the volume integral, all of the terms give zero
when integrated except for φ 0 and the quadratic terms.
〈φ〉 ≈ 1 ∫∫∫ (
a/2
a 3 φ 0 + 1 [
] )
x 2 ∂2 φ
−a/2 2 ∂x 2 + y2 ∂2 φ
∂y 2 + z2 ∂2 φ
∂z 2 dx dy dz (35)
φ=φ 0
〈φ〉 ≈ 1 (
a 3 a 3 φ 0 + a5
24 ∇2 φ ∣ )
φ=φ0
(36)
∇ 2 φ ∣ ∣
φ=φ0
≈ 24
a 2 (〈φ〉 − φ 0) (37)
This approximation gets more accurate as the volume considered for the average
gets smaller. This demonstrates that the Laplacian is a measure of the difference
between the value of the function at the point evaluated and the average value of
the function in a region near that point.
2 Maxwell’s Equations
2.1 Gauss’ Law
∇ · ⃗E = ρ ɛ 0
(38)
The divergence of the electric field is proportional to the charge density at that
point. This means that charge is the source of the electric field. Integrating
over a volume and applying Gauss’ Theorem to Gauss’ Law gives:
∫
∫
∇ · ⃗E ρ
dV = dV (39)
ɛ
∮
0
⃗E · dS ⃗ = Q (40)
ɛ 0
Where Q is the charge enclosed by the surface S. Above is the integral form of
Gauss’ Law. It shows that the flux of the electric field over a closed surface is
proportional to the charge enclosed by that surface.
2.2 Divergence of the Magnetic Field
∇ · ⃗B = 0 (41)
The divergence of the magnetic field is zero, meaning there are no magnetic
monopoles, no sources where the field lines begin or end like the charges for the
electric field. Integrating over a volume and applying Gauss’ Theorem gives:
∫
∫
∇ · ⃗B dV = 0 dV (42)
∮
⃗B · dS ⃗ = 0 (43)
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This integral form shows that the net flux of the magnetic field through a closed
surface is always zero. Every field line that comes in must also leave.
2.3 Faraday’s Law
∇ × E ⃗ = − ∂ B ⃗ (44)
∂t
Integrating over a surface and applying Stokes’ Theorem gives:
∫
∫
∇ × E ⃗ · dS ⃗ ∂B
⃗
= −
∂t · d⃗ S (45)
∮
⃗E · d ⃗ l = − d ∫
⃗B · dS dt
⃗ (46)
Faraday’s Law describes how a change in magnetic flux induces an electric field
on the boundary of the surface considered for the flux. The negative sign in
Faraday’s law is a consequence of Lenz’s Law which states that: the induced
electric field is oriented in such a way as to induce a magnetic field which opposes
the change in magnetic flux.
2.4 Ampere’s Law
∇ × B ⃗ ∂E
= µ 0
⃗j + µ 0 ɛ ⃗ 0 (47)
∂t
Integrating over a surface and applying Stokes’ Theorem gives:
∫
∫ (
∇ × B ⃗ · dS ⃗ ∂
= µ 0
⃗j + µ 0 ɛ ⃗ )
E
0 · dS ∂t
⃗ (48)
∮
∫
⃗B · d ⃗ l = µ 0I ⃗
d + µ0 ɛ 0
⃗E · dS dt
⃗ (49)
Ampere’s Law describes how either currents, or a change in electric flux induces
a magnetic field. The first term, µ 0I, ⃗ in Ampere’s Law show’s how a current
induces a magnetic field that curls around that current according to the righthand-rule.
The second term shows how a change of electric field in free space
creates a magnetic field curling along the boundary of the surface considered for
∂E
electric flux. The quantity ɛ ⃗ 0 ∂t
is called the displacement current, even though
it is not a real current. In current carrying wires, the displacement current
term is usually negligible compared to the first term because the displacement
current includes the very small constant ɛ 0 in addition to µ 0 . We will see later
that this second term is essential to the propagation of light.
2.5 Summary
One can divide Maxwell’s equations into two pairs of equations: the divergence
equations and the curl equations. The divergence equations tell what the static
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sources of the electric and magnetic fields are. The electric field is emitted from
charges, while the magnetic field has no static source.
The curl equations describe how the electric and magnetic fields can be
induced— created by the dynamics of another field (or current) and not from a
static source charge. The induced field is the curled field in each equation with
the source of induction on the right side of each equation.
Maxwell’s equations collected in their entirety:
3 Electrostatics
∇ · ⃗E = ρ ɛ 0
(50)
∇ · ⃗B = 0 (51)
∇ × ⃗ E = − ∂ ⃗ B
∂t
∇ × ⃗ B = µ 0
⃗j + µ 0 ɛ 0
∂ ⃗ E
∂t
3.1 Time-Independent Electric Scalar Potential
(52)
(53)
Charge densities constant in time ⇒ electric fields are constant in time; the
theory of which is called electrostatics. Steady currents ⇒ magnetic fields are
constant in time; the theory of which is called magnetostatics.
In the static senerio, we notice from Faraday’s Law that ∇ × ⃗ E = 0. This
fact can tell us something very unique about the static electric field. It follows
from Stokes’ Theorem that ∮ ⃗ E · d ⃗ l = 0. That is, any line integral of the electric
field around a closed path gives zero. I can break a closed path into two paths
and vary one of them while keeping its end points so that the two paths still
make a closed loop. The line integral of the electric field has to give zero on
this augmented path also. Thus the augmented path must have the same line
integral of the electric field as the original path. In electrostatics, the line integral
of the electric field only depends on the end points of the path. Because of this,
⃗E is called a conservative field. In general, the criterion for a field, ⃗ f, to be
conservative is ∇ × ⃗ f = 0.
Because the line integral of the electric field is independent of path, we can
define a function, φ, called the electric potential:
∫ ⃗r
φ(⃗r) ≡ − ⃗E · d ⃗ l (54)
⃗o
Where ⃗o is the reference point taken to be zero potential. The potential difference
between two points ⃗ b and ⃗a is:
∫ ⃗ b ∫ ⃗a
φ( ⃗ b) − φ(⃗a) = − ⃗E · d ⃗ l + ⃗E · d ⃗ l (55)
⃗o
⃗o
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∫ ⃗ b
= −
= −
⃗o
∫ ⃗ b
⃗a
∫ ⃗o
⃗E · d ⃗ l − ⃗E · d ⃗ l (56)
⃗a
⃗E · d ⃗ l (57)
Which corresponds to the fundamental theorem of gradients that:
φ( ⃗ b) − φ(⃗a) =
∫ ⃗ b
Because this is true for any ⃗a and ⃗ b, it must be that:
⃗a
∇φ · d ⃗ l (58)
⃗E = −∇φ (59)
The fundamental theorem of gradients brings to light the fact that the path
integral of the electric field only depends on the end points because the electric
field can be written as the gradient of some scalar function. The electric potential
is a continuous scalar function in space and the electric field points in the
direction of steepest descent of electric potential. One can also see that writing
the electric field as ⃗ E = −∇φ satisfies the electrostatic case that ∇ × ⃗ E = 0
because the ∇ × ∇f = 0 for any scalar function f.
3.2 Poisson’s Equation
Writing Gauss’ Law with ⃗ E = −∇φ gives:
−∇ · ∇φ = −∇ 2 φ = ρ ɛ 0
(60)
∇ 2 φ = − ρ ɛ 0
(61)
This equation has the form known as Poisson’s Equation:
∇ 2 u = v (62)
With the condition that u → 0 as r → ∞, it has the solution:
u(⃗r) = −1 ∫
v(⃗r ′ )
∣
4π ∣ ∣∣
d 3 r ′ (63)
∣⃗r − ⃗r ′
Therefore the solution to our Poisson’s equation of φ with ∞ chosen to be our
reference point where φ(∞) = 0 is:
φ(⃗r) = 1 ∫
ρ(⃗r ′ )
∣
4πɛ 0
∣ ∣∣
d 3 r ′ (64)
∣⃗r − ⃗r ′
This is a formal solution to Poisson’s Equation. In the context of a problem,
this integral is usually difficult to solve. Later, there is a discussion of more
applicable ways to solve Poisson’s Equation.
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3.3 Coulomb’s Law
Taking minus the gradient of our solution for the electric scalar potential should
give us a general soultion for the electric field for a given charge distribution, ρ.
⃗E(⃗r) = −∇φ(⃗r) (65)
⎛ ⎞
= −1 ∫
ρ(⃗r
4πɛ ′ )∇ ⎝ 1 ∣⎠ 0
∣ ∣∣
d 3 r ′ (66)
∣⃗r − ⃗r ′
)
∫
1 ρ( ⃗r ′ )
(⃗r − ⃗r ′
=
∣
4πɛ 0 ∣ ∣∣
∣⃗r − ⃗r ′ 3
d 3 r ′ (67)
This is known as Coulomb’s Law: that the electric field at position ⃗r generated
by a single particle with charge q at position ⃗r ′ is:
)
⃗E(⃗r) = 1 q
(⃗r − ⃗r ′ ∣
4πɛ 0 ∣ ∣∣
∣⃗r − ⃗r ′ 3
(68)
Therefore, an assortment n of particles creates a field:
⃗E(⃗r) = 1 ∑
n q i
(⃗r − ⃗ )
r i
′ 4πɛ 0 ∣
i=1 ∣⃗r − r ⃗ i
′ ∣ 3 (69)
If the charge distribution is continuous instead of discrete, the sum becomes an
integral over all space:
)
⃗E(⃗r) = 1 ∫ ρ( ⃗r ′ )
(⃗r − ⃗r ′ ∣
4πɛ 0 ∣ ∣∣
∣⃗r − ⃗r ′ 3
d 3 r ′ (70)
The primary importance of fields is that they can be used to determine the
force on a particle. The relation between a force ⃗ F on a particle with charge Q
and the electric field E is:
⃗F = Q ⃗ E (71)
The advantage that taking the intemediate step of calculating the scalar potential
instead of calculating the electric field directly with Coulomb’s law is that
the integral used to calculate the scalar potential is of a scalar quantity while
Coulomb’s law uses vectors. Calculating the scalar potential does not require
the consideration of components. The electric field can easily be determined by
taking minus the gradient of the scalar potential.
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3.4 Electrostatic Energy
3.5 Conductors
3.6 Boundary Conditions of the Electric Field
3.7 Capacitors
3.8 The Uniqueness Theorem
3.9 Methods of Solving Poisson’s Equation
3.9.1 Method of Images
3.9.2 Method of Complex Analysis
3.9.3 Method of Seperation of Variables
4 Magnetostatics
9