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Intermediate Electricity and Magnetism Study Guide
Chapter 2:
Chapter 5:
Coulumb’s Law
Find the Electric Field at a point in space from a known charge configuration or
distribution. The electric field is a function of the charge and the distance away from the charge.
Biot Savart
Find the Magnetic Field at a point in space from a known current or current distribution.
The Magnetic field is a function the current and the distance away from the current (a cross
product).
Gauss’ Law
Find the Electric Field using symmetry in the field due to a known charge configuration or
distribution. To solve these problems we must choose a Gaussian surface where along the surface
the Electric Field is constant due to symmetry. The Electric Field should be parallel or
perpendicular to our Gaussian Surface, or the field could be zero.
Ampere’s Law
Find the Magnetic Field using symmetry (mirror rule) in the field due to a known current
or current distribution. To solve these problems we must choose and Amperian Loop where along
the loop the magnetic field is constant due to symmetry. The amperian loop should be
perpendicular to the magnetic field; it is beneficial if the magnetic field is zero along a side of the
loop.
Electric Potential
Find electric potential from a charge configuration or a distribution using Poisson’s and
Laplace’s equations to break a scenario down to a scalar quantity. This makes it so that you do not
have to deal with components making the problem simpler. From a known electric potential it is
not too difficult to calculate the Electric field, or the charge distribution.
Vector Potential
Find the vector potential from a current distribution. The vector potential is handy because
it is not too difficult to calculate the magnetic field or the current distribution from a known vector
potential.
Energy in a Field
The energy in a field due to a charge distribution can be calculated using the Electric field,
from the electric potential, or a charge configuration based on Coulumb’s Law.
Capacitance
The capacitance is a constant of proportionality, based purely on the geometry of the
problem. Capacitance is useful for determining the Electric Potential, work or energy associated
with the field.
Chapter 3:
Multipole Expansion
When more detail in calculating the electric potential instead of an approximation it can be
beneficial to use a multipole expansion.
Multipole expansion of the vector potential
When more detail in calculating the electric potential instead of an approximation it can be
beneficial to use a multipole expansion.
Chapter 4
Electric Dipoles in an Electric Field
Electric dipole in an electric field becomes polarized. And because of the polarization it
experiences a torque because electric dipoles will seek to align themselves with the electric field.
Induced Dipole
An induced dipole is caused by the separation of the electron and nucleus due to an electric
field. An induced dipole will seek to anti align itself with the electric field.
Magnetic Dipole
The magnetic dipole moment of a loop is a function of the current and the area of the loop.
Polarization
Polarization is the dipole moment per unit volume. A polarized object emits an electric
field.
Magnetization
If an object is placed in a magnetic field the object should become magnetized. When an
object becomes magnetized it experiences a torque because it will seek to align parallel, or
anitparallel with the magnetic field. When an object becomes magnetized it seeks to align itself
somehow due to the magnetic field because of the magnetic dipole moment per unit volume
(magnetization) that occurs. Magnetic dipole moment per unit volume emits a magnetic field.
Bound Charges
It can be easy to calculate the electric potential from a known polarization. The polarization
determines the bound charges; the polarized bound charges emit their own electric field.
Bound Current
The volume and surface bound current can be found by knowing the magnetization. There
is a large amount of bound current within an object. The bound currents emit their own magnetic
field.
Free Charges
Free charges are the charges that are not a result of polarization. The combination of free
and bound charges is the total charge and the enclosed charges will be the sum of both free and
bound charges. Free charges might consist of electrons on a conductor or ions embedded in the
dielectric material or whatever.
Free Current
Free current is the current that is not caused by the magnetic field. The free current is due
to an electric potential difference. The free current distinction is a convenience because then we
do not have to deal with the bound current.
Electric Displacement
D can be calculated from a modification of Gauss’ Law, but is extremely useful because then
we do not need to worry about the bound charges, just the free charges. Knowing the electric
displacement, and the electric field, will tell us the polarization per unit volume.
Auxiliary Field
H permits us to express Ampere’s law in terms of free current. H is extremely useful
because it makes determine the magnetic field die to current much simpler.
Calculating the Electric Field from the Electric Displacement
Calculating the Electric field form the electric displacement is easy because it is a matter of
dividing the electric displacement by the the permittivity of the material. The permittivity of the
material is a function of the electric susceptibility.
Calculating the Magnetic Field from the Auxillary Field
Calculating the magnetic field from the auxillary field is easy because it is a matter of
multiplying the auxillary field by the permeability of the material. The permeability of the material
is a function of the magnetic susceptibility.
EQUATIONS
CHAPTER 2
coulombs law
1 𝑄
𝐸 = 4𝜋𝜀 𝑟 2 𝑟̂
0
Gauss’s Law
𝑄𝑒𝑛𝑐 = ∫ 𝜌 𝑑𝜏
1
𝑑𝑎 = 𝜀 ∫ 𝜌 𝑑𝜏
∮ 𝐸⃑ ∙ ⃑⃑⃑⃑
0
1
⃑⃑⃑⃑ = 𝑄𝑒𝑛𝑐
∮ 𝐸⃑ ∙ 𝑑𝑎
𝜀
0
Stokes’ theorem
∇𝑥𝐸 =0
Potential
𝑟
𝑉(𝑟) ≡ − ∫𝑟𝑒𝑓 𝐸 ∙ 𝑑𝑙
𝐸 = −∇𝑉
1
𝑉(𝑟) = 4𝜋𝜀 ∫
𝜌(𝑟 ′ )
𝑟
0
Energy
𝑊 = 𝑞𝑉
1
𝑊 = 2 ∫ 𝜌𝑉𝑑𝜏
𝑊=
𝑊=
𝜀0
2
𝜀0
2
𝑑𝜏 ′
𝑞
and 𝜌 = 𝑉
∫ 𝐸 2 𝑑𝜏
(∫𝑉 𝐸 2 𝑑𝜏 + ∮𝑠 𝑉𝐸 ∙ 𝑑𝑎)
Capacitance
𝑄
𝐶=𝑉
Force
𝑓 = 𝜌𝐸⃑
𝐹 = ∫ 𝑓 𝑑𝜏
CHAPTER 3
Multipoles
1
1
′ 𝑛
′
𝑉(𝑟) = 4𝜋𝜀 ∑∞
𝑛=0 𝑟 (𝑛+1) ∫(𝑟 ) 𝑃𝑛 (𝑐𝑜𝑠𝜃′)𝜌(𝑟 )𝑑𝜏′
0
1
𝑉𝑚𝑜𝑛 (𝑟) = 4𝜋𝜀
𝑄
0
𝑟
𝜌
𝐸𝑑𝑖𝑝 (𝑟, 𝜃) = 4𝜋𝜀
0𝑟
3
(2𝑐𝑜𝑠𝜃𝑟̂ + 𝑠𝑖𝑛𝜃𝜃̂)
CHAPTER 4
Torque
𝜌 = 𝛼𝐸
𝑁=𝑝𝑥𝐸
𝐹 = (𝑝 ∙ ∇)𝐸
bound charges
𝜎𝑏 = 𝑃 ∙ 𝑛̂
𝜌𝑏 = −∇ ∙ 𝑃
1
𝑉(𝑟) = 4𝜋𝜀 ∮𝑠
𝜎𝑏
0
𝓇
1
𝑑𝑎′ + 4𝜋𝜀 ∫𝑉
0
𝜌𝑏
𝓇
𝑑𝜏′
𝑄𝑡𝑜𝑡 = ∮ 𝜎𝑏 𝑑𝑎 + ∫𝑉 𝜌𝑏 𝑑𝜏
Electric Displacement
𝐷 ≡ 𝜀0 𝐸 + 𝑃
∇ ∙ 𝐷 = 𝜌𝑓
∮ 𝐷 ∙ 𝑑𝑎 = 𝑄𝑓𝑒𝑛𝑐
Linear Dielectrics
𝑃 = 𝜀0 𝜒𝑒 𝐸
𝐷 = 𝜀𝐸
𝜀 = 𝜀0 (1 + 𝜒𝑒 )
1
𝑊 = 2 ∫ 𝐷 ∙ 𝐸 𝑑𝜏
CHAPTER 5
Forces
𝐹𝑚𝑎𝑔 = 𝑄(𝑣 x 𝐵)
𝐹 = 𝑄[𝐸 + (𝑣 x 𝐵)]
𝐹𝑚𝑎𝑔 = ∫ 𝐼(𝑑𝑙 x 𝐵)
bound currents
𝐾 = 𝜎𝑣
𝐽 = 𝜌𝑣
𝐼 = ∫ 𝐽 ∙ 𝑑𝑎 surface
𝐼 = ∫𝑉 (∇ ∙ 𝐽)𝑑𝜏
𝐼
𝐽=𝐴
𝐽 = 𝜌𝑣
Biot-Savart
𝜇
𝐼 x 𝓇̂
𝜇
(𝑑𝑙′ x 𝓇̂)
𝐵(𝑟) = 4𝜋0 ∫ 𝓇 2 𝑑𝑙 ′ = 4𝜋0 𝐼 ∫ 𝓇 2
{for surface and volume currents sub I for K(r’) and dl’ for J(r’) in above eq}
Ampere’s Law
∮ 𝐵 ∙ 𝑑𝑙 ′ = 𝜇0 𝐼𝑒𝑛𝑐
∫(∇ x𝐵) ∙ 𝑑𝑎 = ∮ 𝐵 ∙ 𝑑𝑙 = 𝜇0 ∫ 𝐽 ∙ 𝑑𝑎
∇ x 𝐵 = 𝜇0 𝐽
∇∙𝐵 =0
Vector Potential
𝐵 = ∇x𝐴
∇∙𝐴=0
𝜇
𝐴(𝑟) = 4𝜋0 ∫
𝐽(𝑟 ′ )
𝓇
𝜇0 𝑚 x 𝓇̂
𝐴𝑑𝑖𝑝 = 4𝜋
𝑑𝜏′
𝑟2
Magnetic Dipole Moment
𝑚 = 𝐼 ∫ 𝑑𝑎 = 𝐼𝑎
𝜇 𝑚
𝐵𝑑𝑖𝑝 = 0 3 (2𝑐𝑜𝑠𝜃𝑟̂ + 𝑠𝑖𝑛𝜃𝜃̂)
4𝜋𝑟
CHAPTER 6
Torque
𝑁 = 𝑚x𝐵
Bound Currents
𝐽𝑏 = ∇ x 𝑀
𝐾𝑏 = 𝑀 x 𝑛̂
Ampere’s Law in magnetized materials
1
𝐻 ≡𝜇 𝐵−𝑀
0
∇ x 𝐻 = 𝐽𝑓
∮ 𝐻 ∙ 𝑑𝑙 = 𝐼𝑓𝑒𝑛𝑐
𝐵 = 𝜇0 𝐻
Permeability and Auxiliary Field
𝑀 = 𝜒𝑚 𝐻
𝐵 = 𝜇0 (𝐻 + 𝑀) = 𝜇0 (1 + 𝜒𝑚 )𝐻