Day 13, Wed, Feb 3, Outline
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Potential Diagrams, Representations
Equipotential Surfaces
Electric Potential and Electric Field
Capacitors, Capacitance
Dielectrics, Dielectric Constant
Stored Energy in a Capacitor and in an Electric Field
Capacitors wired in Series and Parallel
Equipotential Surfaces Representations
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Graph of V versus r
Surfaces (3-D view)
Lines on a 2-D (topographic-like) Map
3-D Elevation Diagram
In any/all these cases, the terms equipotential surfaces,
equipotentials, and equipotential lines will be synonymous, since
they’re all representations of the same thing.
Electric Potential of a Point Charge
Slide 21-27
Graphical Representations of Electric Potential
Equipotential Surfaces and the Gradient
• The magnitude of the electric field can be computed from
the potential field:
E=–ΔV/Δs
Δ V = change in potential from one equipotential
surface to another
Δ s = distance between the two equipotential surfaces
Δ V / Δ s = potential gradient = slope!
• Note that the units for the potential gradient must be V/m
and this must also = N/C (units for electric field.)
• Your textbook says something like, “from this point on,
electric field units will be V/m.”
Electric Field as the Potential Gradient
Es = – Δ V / Δ s = – dV / ds
• In multiple dimensions:
Connecting Potential and Field
Slide 21-31
A Conductor in Electrostatic Equilibrium
• Any excess charge must reside on the surface.
• There can be no electric field inside the conductor.
• Electric field outside the conductor is always perpendicular
to the surface.
• The (external) electric field strength is always greatest at the
sharpest or narrowest points.
• But, if the electric field inside is zero, then
E = – Δ V / Δ s = 0, then Δ V = 0.
• It doesn’t say that the potential = 0, but it does say that the
potential must be the same throughout the conductor,
including on its surface.
• The conductor surface must be an equipotential surface.
Capacitors Again
• A capacitor is a device that can store charge. It’s a way to
store energy (electric potential energy.)
• General construction is two conductors separated by some
insulator.
• Two parallel metal plates separated in air form a parallel
plate capacitor.
• Charge is stored electrostatically. The charge is literally
sitting on the plates.
• Originally called a battery, but, a modern battery “stores”
charge/energy chemically rather than electrostatically.
• Also called a condenser. (Rather an old-fashion term now
but still shows up in places.)
Capacitance
• The charge that a capacitor can store on its plates is directly
proportional to the voltage across the plates.
• The constant of proportionality is the capacitance.
Qcapacitor = C ΔVcapacitor
Q = (magnitude of) total charge on (either plate of) capacitor.
(+Q on one plate, –Q on the other plate.)
ΔV = potential difference across the plates
C = the capacitance. The “capacity” to hold charge. A large
capacitance means it can hold a large amount of charge.
Charging a Capacitor
Slide 21-37
Capacitance and Dialectrics
• Rewritten: C = Q / ΔV
Thus, capacitance has units of coulomb/volt (C/V).
1 C/V = 1 farad (F)
• A farad is a lot of capacitance. Commonly measured in
microfarads or picofarads.
• In practice, capacitors are filled with an insulator or
dielectric material.
• The polarization of charge in the dielectric has the effect of
reducing the field, and increasing the capacitance.
• There’s also the practical matter of constructing the devices
in small packages!
Capacitance and Dialectrics (continued)
• The dielectric constant, κ, is the ratio of the field within a
capacitor without the dielectric to the field within the
capacitor with the dielectric:
κ = E / E’
Note that κ is dimensionless, unitless.
κ is a number greater than (or equal to) 1.
= 1 for vacuum. (Close to 1 for air.)
E = field (in capacitor) without dielectric
E’ = field (in capacitor) with dielectric
Dielectrics and Capacitors
Slide 21-39
Capacitance (continued)
• Remember, for a parallel plate capacitor
Ecapacitor = Q / ε0 A
• But E is also equal to the gradiant: E = Δ V / Δ s
• For a capacitor, Δ V = VC when
Δ s = distance between plates = d
So:
Q / ε 0 A = VC / d
Q = ε0 A VC / d
• Thus, the capacitance is also equal to:
C = ε0 A / d
(for air or vacuum filled)
C = κ ε0 A / d
(with dieletric)
Energy Stored in a Capacitor
• In a charged capacitor, we’ve got charges at some potential.
That means there’s potential energy!
• The energy stored in a capacitor:
UC = ½ C VC2 = ½ Q2 / C
and also:
= ½ (κ ε0 A / d) (E d)2
= ½ (κ ε0 A d) E2
• But A d is = volume, thus
Energy density = Energy / Volume = UC / A d
= ½ κ ε0 E2
• True for any electric field, whether inside a capacitor or not.
Capacitors in Parallel Combinations
• Capacitors wired in parallel will all see the same potential
difference across their plates, so they’ll all store up charge
= C ΔVC and the total charge stored will be the sum of the
individual capacitor’s charges.
• But, the total charge stored divided by the potential ΔVC
will then be the equivalent capacitance.
• The equivalent capacitance for capacitors in parallel is:
Ceq = C1 + C2 + C3 + … Cn
Capacitors in Series Combinations
• Capacitors wired in series will have the same charge on
each of them. The potential difference across each will be
dependent on their capacitances. VC1 = Q / C1
• But, the total potential across all capacitors divided by the
charge stored will then be the equivalent capacitance.
• The equivalent capacitance for capacitors in series is:
Ceq = (1/C1 + 1/C2 + 1/C3 + … 1/Cn )-1
• If you know the rules for resistors in series and parallel
combinations already, you should note that the forms of
these equations are opposite those for resistors.