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PHAS0038 BACKGROUND INFORMATION FOR EXAMINATION
PHYSICAL CONSTANTS
The following data may be used if required:
Speed of light in vacuum, c = 2.998 × 10
8 m · s
− 1
Permittivity of free space, 0 = 8.85 × 10
− 12 F · m
− 1
Permeability of free space, μ 0 = 4π × 10
− 7 H · m
− 1
Fundamental magnitude of electronic charge, e = 1.602 × 10
− 19 C
Mass of an electron, me = 9.11 × 10
− 31 kg
VECTOR IDENTITIES
In the formulae below, F , G and C denote vector fields, and φ and ψ are scalar functions.
C × ( F × G ) = ( C · G ) F − ( C · F ) G (1)
∇ · ∇φ = ∇
2 φ (2)
∇ · ∇ × F = 0 (3)
∇ × ∇φ = 0 (4)
∇ × (∇ × F ) = ∇(∇ · F ) − ∇
2 F (5)
∇(φψ) = (∇φ)ψ + φ(∇ψ) (6)
∇(φ F ) = (∇φ) · F + φ∇ · F (7)
∇ · ( F × G ) = (∇ × F ) · G − (∇ × G ) · F (8)
∇ × (φ F ) = (∇φ) × F + φ∇ × F (9)
∇ × ( F × G ) = (∇ · G ) F − (∇ · F ) G + ( G · ∇) F − ( F · ∇) G (10)
VECTOR OPERATORS IN DIFFERENT COORDINATE SYSTEMS
- Cartesian : Coordinates (x, y, z), Volume element dV = dx dy dz
∇ψ = i x
∂ψ
∂x
∂ψ
∂y
∂ψ
∂z
(11a)
∇ · F =
∂Fx
∂x
∂Fy
∂y
∂Fz
∂z
(11b)
∇ × F =
i x i y i z
∂ ∂x
∂ ∂y
∂ ∂z
Fx Fy Fz
(11c)
2 ψ =
2 Fx
∂x^2
2 Fy
∂y^2
2 Fz
∂z^2
(11d)
- Cylindrical : Coordinates (R, ϕ, z), Volume element dV = R dR dϕ dz
∇ψ = i R
∂ψ
∂R
R
∂ψ
∂ϕ
∂ψ
∂z
(12a)
∇ · F =
R
∂(RFR )
∂R
R
∂Fϕ
∂ϕ
∂Fz
∂z
(12b)
∇ × F =
R
i R R i ϕ i z
∂ ∂R
∂ ∂ϕ
∂ ∂z
FR RFϕ Fz
(12c)
2 ψ =
R
∂R
R
∂ψ
∂R
R
2
2 ψ
∂ϕ
2
2 ψ
∂z
2
(12d)
- Spherical polar : Coordinates (r , θ, ϕ), Volume element dV = r
2 sin θ dr dθ dϕ
∇ψ = i r
∂ψ
∂r
r
∂ψ
∂θ
r sin θ
∂ψ
∂ϕ
(13a)
∇ · F =
r 2
∂(r
2 Fr )
∂r
r sin θ
∂(sin θFθ)
∂θ
r sin θ
∂Fϕ
∂ϕ
(13b)
∇ × F =
r 2 sin θ
i r r i θ r sin θ i ϕ
∂ ∂r
∂ ∂θ
∂ ∂ϕ
Fr r Fθ r sin θFϕ
(13c)
2 ψ =
r 2
∂r
r
2 ∂ψ
∂r
r 2 sin θ
∂θ
sin θ
∂ψ
∂θ
r
2 sin
2 θ
2 ψ
∂ϕ^2
(13d)
IMPORTANT THEOREMS
- Stokes’ Theorem equates the integral of the curl of a vector function F , over an open
surface S, to the line integral of that same function around the boundary (closed curve
Γ ) of that surface:
S
∇ × F · dS =
Γ
F · d `. (14)
- Divergence Theorem equates the integral of a vector function G over a closed surface
S, to the volume integral of the divergence of that same function over the volume V
enclosed by that surface:
V
∇ · G dV =
S
G · n dS, (15)
where n denotes a unit vector locally normal to the surface.
PHAS0038/2022-23 CONTINUED
TIME-DEPENDENT POTENTIALS AND FIELDS
A is the vector potential, ϕ is the scalar potential.
B = ∇ × A , (21)
E = −∇ϕ −
∂ A
∂t
BIOT-SAVART LAW (MAGNETIC FIELD FROM A STATIC CURRENT DISTRIBUTION)
Contribution dB to the magnetic field at the test point position r , from a current element
situated at the source position r
′ , is:
dB =
μ 0
4 π
I
′ dl
′ × ( r − r
′ )
| r − r
′ |^3
where I
′ is current and dl
′ is an infinitesimal vector in the direction of current flow.
MAGNETIC POTENTIAL FROM A TIME-VARYING CURRENT DISTRIBUTION
A ( r , t) =
μ 0
4 π
V′
J ( r
′ , tr )
| r − r
′ |
dV
′ , (24)
where symbols have their usual meaning, and the primed quantities are related to the volume
integral of a source distribution of current density. The quantity tr = t −| r − r
′ |/c is the retarded
time, related to the time required for light to propagate over the distance between the source
point r
′ and the test point r.
LOCAL RADIATION AT A MOVING CHARGED PARTICLE
Electromagnetic power radiated (to infinity) per unit solid angle from a moving particle
(point charge):
dP
dΩ
q
2
16 π
2 0
| r ˆ × ( u × a )|
2
( r ˆ · u )
5
where q is particle charge, r ˆ is a unit vector in the direction of the radiation, a is particle
acceleration, v is particle velocity, u = c rˆ − v.
PHAS0038/2022-23 CONTINUED
SPECIAL RELATIVISTIC TRANSFORMATIONS
Inertial coordinate frames S(x, y, z, t) and S
′ (x
′ , y
′ , z
′ , t
′ ) are defined to have Cartesian
coordinate systems, aligned in such a way that the origins and axes of both frames coincide
at time zero within both frames (i.e. when t = t
′ = 0); and the frame S
′ moves with velocity v
in the x direction, as seen by S.
The Lorentz coordinate transformations are:
x
′ = γ (x − v t) (26a)
y
′ = y (26b)
z
′ = z (26c)
t
′ = γ (t − v x/c
2 ), (26d)
where γ = 1/
1 − v 2 /c^2.
The spacetime coordinate interval between two events ∆x
2
2
2 − c
2 ∆t
2 is invariant
under the Lorentz transformations.
The field transformation equations for each Cartesian component of the electric and
magnetic fields are:
E
′ x =^ Ex^ ,^ E
′ y =^ γ(Ey^ −^ v Bz^ ),^ E
′ z =^ γ(Ez^ +^ v By^ )
B
′ x =^ Bx^ ,^ B
′ y =^ γ(By^ +^
v
c^2
Ez ), B
′ z =^ γ(Bz^ −^
v
c^2
Ey ),
where γ = 1/
1 − v
2 /c
2 , and c is the speed of light in vacuum.
END OF PHAS0038 EXAMINATION BOOKLET
