Exam 1 Review Lecture: Chs. 22-26
Ch. 22: Electric Charges






Matter is comprised of electric particles: all matter made of atoms
o Atoms composed of protons (+) and neutrons (neutral) in the nucleus, surrounded by a
cloud of electrons (-)
o Charge on proton exactly equal to charge on electron: 𝑒 = 1.6 × 10−19 𝐶
Like charges repel (+ repels +, - repels -), unlike charges attract (+ attracts –, and – attracts +)
Insulators are materials in which all of the electrons are bound to atoms
Metal conductors have a small fraction (~ 1/1013 ) of the electrons within the substance which
are not bound to atoms, but are free to move
Both insulators and conductors can become polarized
o In conductors, the free charges move to one end of the conductor
o In insulators, the electrons in individual atoms spend more time on one side of the
nucleus than the other, causing polarization at the atomic level
 Two opposite charges with a slight separation between them form an electric
dipole. Atoms within a polarized material are dipoles. Many common
molecules are also dipoles (e.g., water).
Coulomb’s Law: force between two point charges
𝑘 |𝑞1 | |𝑞2 | |𝑞1 ||𝑞2 |
=
|𝐹1𝐸𝑜𝑛 2 | = |𝐹2𝐸𝑜𝑛 1 | =
𝑟2
4𝜋𝜖0 𝑟 2
where|𝑞1 | and |𝑞2 | are the magnitudes of the charges, and 𝑟 is the straight-line distance
between the charges
o Force is attractive if the charges are opposite in sign, and repulsive if they are the same
1
o
k = 4 𝜋 𝜖 = 8.99 × 109 𝑁 𝑚2 ⁄𝐶 2 (SI Units) (Coulomb’s constant)
o
𝜖0 ≈ 8.85 × 10−12 𝐶 2 ⁄𝑁 𝑚2 (permittivity of free space)
Coulomb’s Law follows superposition principle: if multiple charges present, can sum
forces between individual pairs to get net force
0
⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
∑ ⃗⃗⃗
𝐹1 = ⃗⃗⃗⃗⃗⃗
𝐹21 + 𝐹
31 + 𝐹41 + ⋯

Space around an electrically charged object is filled with electric field
o Electric field is a vector (𝐸⃗ )
o SI units of N/C (sometimes also expressed as V/m)
o Magnitude of electric field for a point charge:
𝑘𝑞
𝐸= 2
𝑟
o Direction of electric field for a point charge:

Away from a positive source charge, towards a negative source charge


These lines of electric field are in the radial direction:
𝑘𝑄
𝑄
𝐸⃗ = 2 𝑟̂ =
𝑟̂
𝑟
4𝜋𝜖0 𝑟 2
The Coulomb force between two point charges can then be thought of as the electric field due
to one charged particle interacting with another charged particle:
𝑘𝑞𝑄
𝑞𝑄
𝐹 = 2 𝑟̂ =
𝑟̂
𝑟
4𝜋𝜖0 𝑟 2
𝐹 = 𝑞𝐸⃗
Ch. 23: Electric Field


Representations of the electric field
o Besides drawing representative vectors at various locations, can also draw electric field
lines:
 Continuous curves tangent to electric field vectors
 Density of field lines indicates strength of field (more closely spaced lines =
greater field strength)
 Electric field lines start on positive charges and end on negative charges
 Electric field lines never cross
Dipoles
o Orientation of a dipole defined by dipole moment (𝑝 = 𝑞𝑠, where 𝑠 is the separation
distance between the charges): vector pointing from center of negative charge to center
of positive charge
o
o
On the axis of a dipole (along the line connecting the charges):
2𝑘𝑝
𝐸⃗𝑑𝑖𝑝𝑜𝑙𝑒, ≈ 3 , 𝑜𝑛 𝑡ℎ𝑒 𝑎𝑥𝑖𝑠 𝑜𝑓 𝑎 𝑑𝑖𝑝𝑜𝑙𝑒
𝑟
On the line bisecting a dipole (perpendicular to the axis):
−𝑘𝑝
, 𝑜𝑛 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑖𝑝𝑜𝑙𝑒
𝑟3
 From close up, you see the electric field of the dipole. As you move farther from the
dipole, the particle separation appears smaller and smaller: only see the net charge
of zero (and hence, no electric field)
Like electric forces, electric fields also follow superposition principle: can sum the fields of
individual particles to find the net field at a given location
o For large, continuous objects, can break down into infinitesimally small point charges
(𝑑𝑞𝑠 ), and then sum over all 𝑑𝑞𝑠  for small enough 𝑑𝑞𝑠 , this becomes an integral:
𝑑𝑞
⃗⃗⃗⃗𝑠 = 𝑘 ∫ 𝑠 𝑟̂
𝐸⃗ = ∫ 𝑑𝐸
2 𝑠𝑃
𝑟𝑠𝑃
where 𝑟̂
𝑠𝑃 is the vector from element s within object to point of interest P
o To evaluate integral, need to express charge in terms of charge density: charge per unit
length, area, or volume. Then integrate over length/area/volume to get total charge.
𝑄
𝑄
𝑄
𝜆= ,
𝜂= ,
𝜌=
𝐿
𝐴
𝑉
Useful results: electric fields due to some symmetric continuous charge distributions
o Electric field a distance 𝑟 from a line of charge of length 𝐿 (thin rod, wire, etc.), along the
line bisecting the line of charge:
𝑘|𝑄|
𝑎𝑤𝑎𝑦 𝑓𝑟𝑜𝑚 𝑙𝑖𝑛𝑒 𝑖𝑓 +
𝐸⃗𝑙𝑖𝑛𝑒,𝑓𝑖𝑛𝑖𝑡𝑒 =
,{
}
2
𝑡𝑜𝑤𝑎𝑟𝑑𝑠 𝑙𝑖𝑛𝑒 𝑖𝑓 −
𝐿
2
𝑟 √𝑟 + ( )
2
direction is away from rod if positively charged, towards rod if negatively charged
o Electric field a distance 𝑟 away from an infinite line of charge:
2𝑘|𝜆| 𝑎𝑤𝑎𝑦 𝑓𝑟𝑜𝑚 𝑙𝑖𝑛𝑒 𝑖𝑓 +
𝐸⃗𝑙𝑖𝑛𝑒,𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 =
,{
}
𝑡𝑜𝑤𝑎𝑟𝑑𝑠 𝑙𝑖𝑛𝑒 𝑖𝑓 −
𝑟
o Electric field on the axis of a ring of charge of radius 𝑅 and total charge 𝑄:
𝑘𝑄𝑧
𝐸⃗𝑟𝑖𝑛𝑔,𝑧 = 2
(𝑧 + 𝑅 2 )3⁄2
o Electric field on the axis of a disk of radius 𝑅 and total charge 𝑄:
𝜂
𝑧
𝐸⃗𝑑𝑖𝑠𝑘,𝑧 =
[1 −
]
2𝜖0
√𝑧 2 + 𝑅 2
o Electric field due to an infinite plane (sheet) of charge:
𝜂 𝑎𝑤𝑎𝑦 𝑓𝑟𝑜𝑚 𝑝𝑙𝑎𝑛𝑒 𝑖𝑓 +
𝐸⃗𝑝𝑙𝑎𝑛𝑒 =
,{
}
𝑡𝑜𝑤𝑎𝑟𝑑𝑠 𝑝𝑙𝑎𝑛𝑒 𝑖𝑓 −
2𝜖0
o Electric field a distance 𝑟 from the center of a sphere of radius 𝑅 and total charge 𝑄,
where 𝑟 ≥ 𝑅:
𝑘𝑄
𝐸⃗𝑠𝑝ℎ𝑒𝑟𝑒 = 2 𝑟̂ ,
𝑓𝑜𝑟 𝑟 ≥ 𝑅
𝑟
Parallel-plate capacitor
o Electric field between the plates:
𝜂
𝐸⃗𝑖𝑛𝑠𝑖𝑑𝑒 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟 = , 𝑓𝑟𝑜𝑚 + 𝑡𝑜 − 𝑝𝑙𝑎𝑡𝑒
𝜖0
o Electric field outside the plates:
𝐸⃗𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟 = 0
𝐸⃗𝑑𝑖𝑝𝑜𝑙𝑒 ≈





Motion of a charged particle in an electric field
o An external electric field exerts a force on a charged particle, causing it to accelerate:
𝐹 = 𝑞𝐸⃗
𝑚𝑎 = 𝑞𝐸⃗
𝑞
𝑎 = 𝐸⃗
𝑚
o If the electric field is uniform (constant in magnitude and direction at all points), then
the acceleration of the charged particle is constant, and constant-acceleration
kinematics equations apply
Motion of a dipole in an electric field
o In a uniform electric field, net force is zero, but there is a torque
o Torque rotates dipole to align dipole moment with direction of electric field: oscillates
around aligned position; with enough damping, comes to rest in aligned position
𝜏𝑛𝑒𝑡 = 𝑝 × 𝐸⃗
𝜏𝑛𝑒𝑡 = 𝑝𝐸 sin 𝜃
𝜏𝑛𝑒𝑡 = 𝑞𝑠𝐸 sin 𝜃
Ch. 24: Gauss’s Law
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
Three major types of symmetry: planar, cylindrical, and spherical
o The symmetry of the electric field will match the symmetry of the charge distribution
 Allows us to rule out many possible field shapes, which allows us to apply
mathematical tools to the single remaining possible shape
Electric flux: amount of electric field passing through a given surface
𝛷𝐸 = 𝐸⃗ ∙ 𝐴 𝑛̂ = 𝐸⃗ ∙ 𝐴 = 𝐸𝐴 cos 𝜃
o 𝐸⃗ is the electric field vector
o 𝐴 is the area of the surface
o 𝑛̂ is the unit vector normal (perpendicular) to the surface at any given point
o 𝐴 = 𝐴 ̂𝑛 is the area vector perpendicular to the surface at any given point
o 𝜃 is the angle between vectors 𝐸⃗ and 𝐴
 Note: Flux is a scalar
2
Units: 𝑁𝑚 ⁄𝐶
o Net flux through a closed surface enclosing no charge is zero

Gauss’s Law:
𝛷𝐸 = ∮ 𝐸⃗ ∙ 𝑑𝐴 =

𝑞𝑒𝑛𝑐
𝜖0
Choose an imaginary closed surface (the Gaussian surface) which includes the point of interest
𝑞
on the surface. Then, the integral over that entire closed surface of 𝐸⃗ ∙ 𝑑𝐴 is equal to 𝜖𝑒𝑛𝑐 ,
0
where 𝑞𝑒𝑛𝑐 is the total charge enclosed by that surface
o 𝑞𝑒𝑛𝑐 may be:
 the sum of all point charges enclosed
 a uniform charge density multiplied by the total volume enclosed by the
Gaussian surface
 a non-uniform charge density integrated over the total volume enclosed by the
Gaussian surface
o While Gauss’s Law is true for any shape of closed surface surrounding any charge shape
of charge distribution, exact mathematical solutions only exist for cases where
symmetry exists
 Can be spherical, cylindrical, or planar symmetry
 Choose a Gaussian surface that reflects the symmetry of the given charge
distribution
 This allows the angle between 𝐸⃗ and 𝑑𝐴 to be either 0° or 90° through
all faces of the Gaussian surface
o 𝐸 𝑑𝐴 cos(0°) = 𝐸 𝑑𝐴
o 𝐸 𝑑𝐴 cos(90°) = 0
 Then, ∮ 𝐸⃗ ∙ 𝑑𝐴 only has non-zero contributions from the portions of the
surface where the angle between 𝐸⃗ and 𝑑𝐴 is 0°
 Since 𝐸 is independent of 𝑑𝐴, this integral becomes 𝐸 ∮ 𝑑𝐴
o The integral of dA over a closed surface simply produces the
equation for the area A of that surface
o Often, you do not need to actually integrate your Gaussian
surface:
∮ 𝐸⃗ ∙ 𝑑𝐴 = 𝐸 ∗
[𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑎𝑟𝑡 𝑜𝑓 𝐺𝑎𝑢𝑠𝑠𝑖𝑎𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑤ℎ𝑒𝑟𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐸⃗ 𝑎𝑛𝑑 𝑑𝐴 𝑖𝑠 0°]
o Charges do not have to be at center of volume enclosed by surface
o Also works for multiple charges, or any shape of distribution of charges: can add electric
fields vectorially, so can take net 𝐸⃗ to find flux

Useful symmetric results:
o Solid non-conducting spherical shell of uniformly distributed total charge 𝑄 and radius 𝑅
 Gaussian surface is a sphere
o

⃗
Inside shell: 𝐸

⃗
Outside shell: 𝐸
=
𝑄𝑟
4𝜋𝜖0 𝑅 3
𝑄
=
𝑟̂ , 𝑟 < 𝑅
4𝜋𝜖0 𝑟 2
𝑟̂ , 𝑟 > 𝑅
Hollow conducting spherical shell carrying charge Q
o


Gaussian surface is a sphere
Inside hollow cavity of shell: 𝐸⃗ = 0

⃗
Outside spherical shell: 𝐸
𝑘𝑄
𝑟2
𝑟̂
(from outside, looks like a point charge)
Infinite one-dimensional sheet of uniform charge density 𝜂
 Gaussian surface is a cylinder


=
⃗
On each side of sheet, 𝐸
=
𝜂
2 𝜖0
in direction perpendicular to surface of sheet
Conductors
o Conductors in electrostatic equilibrium (i.e., no moving charges) carry all of their
excess charge at their exterior surfaces. Therefore, within the material of the
conductor itself, the electric field is zero.
o At the exterior surface of the conductor (where the charge is located), the electric field
must be perpendicular to the surface at that point, and its magnitude is
𝜂
𝐸𝑠𝑢𝑟𝑓𝑎𝑐𝑒 =
𝜖0
o The shape of the conductor may include an interior surface (i.e., a hole).
 If the hole contains no charge, the interior surface has no charge, and the hole
contains no electric field
 This is how a Faraday cage works! You can place a neutral conducting
box inside a region of electric field to exclude electric field from the area
inside the box – process called screening.
 If the hole has charge within it, there is an opposite induced charge on the
interior surface of the conductor and an electric field within the space of the
hole (but still not within the material of the conductor)
Ch. 25: Electric Potential Energy and Electric Potential

Imagine two positive point charges a distance 𝑅 apart. They want to repel, so it requires work
to bring them together (imagine putting pressure on a spring connecting them). That work is
the electrostatic potential energy (𝑈𝐸 ).
o Place one charge “for free” (no potential energy associated with it). The work required
to bring in a second charge from ∞ is
∞
𝑊𝑎𝑝𝑝 = ∫ 𝐹𝐸 ∙ 𝑑𝑟
𝑅
∞
𝑘 𝑞𝑞 ′
𝑑𝑟
𝑟2
𝑅
𝑘 𝑞𝑞 ′
=
= 𝑈𝐸
𝑅
𝑊𝑎𝑝𝑝 = ∫
𝑊𝑎𝑝𝑝






 𝑈𝐸 is a scalar
 SI units of Joules (J)
o If both charges are positive or both are negative, do positive work to bring them
together (charges want to repel, takes work to bring them together)
o If one positive and one negative, do negative work to bring them together (charges
want to attract, applied force must be opposite direction of motion to keep them from
coming together)
Does work depend on path taken to get from ∞ to 𝑅? No  electric force is conservative!
Net work done to assemble a collection of charges
o Can bring each charge in from infinity and sum up work for each charge. Can do this for
positive or negative charges (some work will be positive, some will be negative). This
net work is equivalent to 𝑈𝐸 .
Potential energy of a dipole
𝑈𝐸,𝑑𝑖𝑝𝑜𝑙𝑒 = −𝑝 ∙ 𝐸⃗
𝑈𝐸,𝑑𝑖𝑝𝑜𝑙𝑒 = −𝑝𝐸 cos 𝜑
Electric potential: work per unit charge done by an applied force to assemble a collection of
charges from infinity. For a point charge,
𝑈𝐸
𝑘𝑞
𝑞
𝑉= ′ =
=
𝑞
𝑟
4𝜋𝜖0 𝑟
o Depends on “source charge” (𝑞, first charge or charge distribution present) only; no
dependence on “test charge(s)” (𝑞 ′ , other charge(s) brought in from infinity)
o Scalar
o SI Units: J/C = Volts (V)
o Can be positive or negative, depending on sign of originating charge
Equipotential surfaces are mathematical surfaces with the same value of 𝑉 at every point. For a
point charge, they are concentric spheres.
o Electric field lines are always perpendicular to equipotential surfaces
o Equipotential surfaces with different values can never intersect
Electric potential also gives us a new way to define the electric potential energy of a point
charge in a region of electric potential:
𝑈𝐸 = 𝑞 ′ 𝑉
o Important: 𝑈𝐸 is not the same as 𝑉! For a given collection of charges, there is a
single value of 𝑈𝐸 : the amount of work required to assemble that collection of
charges. However, 𝑉 will be different at different points in space near the
collection of charges




Usually, we discuss the electric potential difference, or simply the potential difference between
two points in space.
o Positive charges want to travel from regions of high electric potential to regions of low
electric potential.
o For negative charges, it’s the opposite: want to go from regions of low electric potential
to regions of high electric potential.
o So far, we have mathematically defined 𝑉 = 0 to be at ∞ because of 1⁄𝑟 dependence.
However, as with gravitational potential energy, can choose your zero point to be
anywhere convenient because you’re only interested in the difference between two
points. Electrical engineers typically choose the Earth to be at zero electric potential –
hence, the term “ground”.
What about multiple charges? Can add the potentials at a given location from each charge
individually, as if the other charges were not present
o Negative charges have negative V, and positive charges have positive V, so net V can be
positive, negative, or zero for a given point in space
o A reminder: E = 0 does NOT mean V = 0. Rather, it means V = constant at that point in
space. Likewise, V = 0 does NOT mean E = 0. It means that bringing a charge from
infinity to that location requires zero work.
Electric potential inside a parallel plate capacitor with plate separation distance 𝑑:
∆𝑉𝐶 = 𝐸𝑑
At a distance 𝑠 < 𝑑 from the negative plate,
𝑠
𝑉 = ∆𝑉𝐶
𝑑
Electric potential of a uniformly positively charged sphere (+𝑄) of radius 𝑅
o Outside the sphere:
𝑘𝑄
𝑉=
,
𝑟>𝑅
𝑟
 This can also be expressed as
𝑅
𝑉 = 𝑉0
𝑟
where 𝑉0
=
𝑘𝑄
𝑅
is the electric potential at the surface of the sphere
Ch. 26: Electric Potential and Electric Field

If the electric field in the region between two locations is known, you can calculate the potential
difference between those points:
𝑓
∆𝑉 = − ∫ 𝐸⃗ ∙ 𝑑𝑠
𝑖
o




Note: minus sign indicates that the magnitude of the potential decreases along the
direction of the field lines.
o Can calculate integral, or can solve graphically:
∆𝑉 = −(𝐴𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝐸𝑠 𝑣𝑠. 𝑠 𝑐𝑢𝑟𝑣𝑒)
𝑉𝑓 = 𝑉𝑖 − (𝐴𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝐸𝑠 𝑣𝑠. 𝑠 𝑐𝑢𝑟𝑣𝑒)
Can take the inverse operation to find the electric field from the electric potential:
⃗𝑉
𝐸⃗ = −∇
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝐸⃗ = − ( 𝑖̂ +
𝑗̂ +
𝑘̂ )
𝜕𝑥
𝜕𝑦
𝜕𝑧
Equipotential surfaces and Kirchoff’s Loop Law
o Work done in moving a charge from point A to point B is independent of the path. So,
potential difference between point A and point B is also independent of the path.
o If path is a closed loop, ∆𝑉𝑙𝑜𝑜𝑝 = ∑𝑖(∆𝑉)𝑖 = 0, where the path is made up of 𝑖
segments.
Conductors in electrostatic equilibrium
o Any two points within the material of the conductor are at the same electric potential,
including the surface.
o Since electric fields must be always perpendicular to equipotential surfaces, we can
conclude that external electric fields must always be perpendicular to the surface of a
charged conductor.
 Equipotential surfaces near electrodes must roughly match the shape of the
electrode.
 The surface charge density, and therefore the electric field strength, are
greatest at “sharp” corners.
Separation of charges simultaneously creates electric field in the space between the charges,
and produces a potential difference between their locations.
o Electric forces try to bring positive and negative charges together, so we need a
nonelectric mechanism to separate the charges.
 Van de Graaff generator: mechanical separation of charges
 Battery: chemical separation of charges
o Batteries: can use the “charge escalator” model of a battery
 Work is done by chemical reactions in moving positive charges from the
negative terminal of the battery to the positive terminal
 The emf of the battery is the work done per unit charge:
𝑊
𝜀=
𝑞



“Electromotive force” is a misnomer! Not a force, but rather, a
potential difference. Usually referred to as “emf” to avoid confusion.
o This separation of charges produces a potential difference ∆𝑉 between the terminals
 For an ideal battery, ∆𝑉 = 𝜀
 If current is flowing through the battery (realistic situation), then ∆𝑉 < 𝜀
Capacitor: a system for storing electric potential energy using two conductors
o Capacitance is the capacity of an object to hold charge at a given electric potential
difference:
𝑄
𝐶=
∆𝑉𝐶
where 𝑄 is the maximum charge on one conductor and ∆𝑉𝐶 is the potential difference
between the conductors.
 Units: Farads (F)
1 Farad = 1 Coulomb/1 Volt (F = C/V) (named after Faraday)
 Each conductor in the pair holds the same maximum amount of charge, but one
holds positive charge and the other holds negative charge
o Parallel-plate capacitor:
𝜖0 𝐴
𝐶=
𝑑
Circuits and capacitors
o 2 ways to connect circuit elements: in parallel or in series


Elements wired in parallel have the same potential difference across them (∆𝑉
is the same for each branch of the circuit), but may have different current in
each branch
 Elements wired in series have the same current flowing through them, but may
have different ∆𝑉 across them. For capacitors, this means that capacitors in
series within a circuit will hold the same charge.
o Often, circuits can be very complex  easier to analyze a simpler, but electrically
identical, circuit. To do this, we need to calculate the equivalent circuit elements.
 For capacitors wired in parallel:
𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + 𝐶3 + 𝐶4 + ⋯
 For capacitors wired in series:
1
1
1
1
1
= + + + +⋯
𝐶𝑒𝑞 𝐶1 𝐶2 𝐶3 𝐶4
Energy stored in a capacitor:
1
1
𝑈 = 𝑄∆𝑉𝐶 = 𝐶 (∆𝑉𝐶 )2
2
2
o

Note: an isolated capacitor (not connected to a circuit) has a constant value of Q. For a
capacitor connected to a battery, the potential difference across the capacitor is
constant, not the charge.
Capacitors with dielectrics
o A dielectric is a non-conducting (insulating) material often placed between the plates of
a capacitor
o Electric field of the capacitor causes polarization of the atoms/molecules within the
dielectric material (or, if they’re already polarized, causes the atoms/molecules to align
with the electric field)  produces a secondary, induced electric field within the
dielectric, opposite the direction of the electric field from the capacitor, reducing the
electric field strength inside
o Define the dielectric constant as
𝐸0
𝜅≡
𝐸
 The electric field inside the capacitor is reduced by a factor of κ:
𝐸0
𝐸=
𝜅
 For an isolated capacitor (not connected to a battery  𝑄 is constant)
 The electric potential inside the capacitor is reduced by a factor of κ:
(∆𝑉𝐶 )0
∆𝑉𝐶 =
𝜅
 For a capacitor connected to a battery, ∆𝑉𝐶 will remain constant, and more
charge will flow to the capacitor:
𝑄 = 𝜅𝑄0
 In either situation, the capacitance is increased by a factor of κ:
𝐶 = 𝜅𝐶0