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Unformatted text preview: Capacitance and Dielectrics
Chapter Outline ENGINEERING PHYSICS II
Capacitance
PHY 303L Capacitors in Combination
Dielectrics Chapter 26: Capacitance and Dielectrics Energy in Capacitors Maxim Tsoi
Physics Department,
The University of Texas at Austin http://www.ph.utexas.edu/~tsoi
303L: Capacitance and Dielectrics (Ch.26) 303L: Capacitance and Dielectrics (Ch.26) Capacitance Calculating Capacitance Definition • Capacitor Single conductor combination of two conductors carrying 1) Assume a charge Q on conductors charges of equal magnitude and opposite sign
2) Calculate the potential difference
• Capacitance ratio of the magnitude of the charge
3) Use C=Q/∆V to evaluate the capacitance on either conductor to the magnitude of the potential
difference between the conductors C= • Capacitance of a spherical charged single Q
∆V conductor: C= • C is a measure of a capacitor’s ability to store charge
• C is always a positive quantity Q
Q
R
=
=
∆V k e Q R k e • Capacitance of a pair of conductors depends • The SI unit of C is farad (F): 1 F = 1 C/V; on the geometry of the conductors • Typical devices have C ranging from µF to pF 303L: Capacitance and Dielectrics (Ch.26) 303L: Capacitance and Dielectrics (Ch.26) Calculating Capacitance Electric Circuits Parallelplate capacitors Circuit diagrams Two parallel metallic plates of equal area A are separated
by a distance d and carry charges Q and –Q, respectively
• The electric field
between the plates
• The potential difference
between the plates E= Circuit symbols Circuit diagrams Q
σ
=
ε0 ε0A ∆V = Ed = Qd ε0A Capacitance of a parallelplate capacitor is proportional to the
area of its plates and inversely proportional to the plate separation C= 303L: Capacitance and Dielectrics (Ch.26) Q
Q
εA
=
=0
∆V Qd ε 0 A
d 303L: Capacitance and Dielectrics (Ch.26) 1 Combinations of Capacitors Combinations of Capacitors Parallel combination Series combination • Individual potential differences across capacitors connected in parallel are the • The charges on capacitors connected in series are the same same and are equal to the potential difference applied across the combination ∆V = ∆V1 + ∆ V2 = Q C eq Q = Q1 + Q2
Q1 = C1 ∆V C eq = ∆V1 = Q C1 Q
= C1 + C 2
∆V 1111
= + + + ...
Ceq C1 C2 C3 Ceq = C1 + C2 + C3 + ...
The equivalent capacitance of a parallel combination of
capacitors is the algebraic sum of the individual capacitances ∆ V2 = Q C 2 1
1
1
=
+
C eq C1 C 2 Q 2 = C 2 ∆V The inverse of the equivalent capacitance of a series combination of
capacitors is the algebraic sum of the inverses of the individual capacitances 303L: Capacitance and Dielectrics (Ch.26) 303L: Capacitance and Dielectrics (Ch.26) Energy Stored in a Charged Capacitor Energy Stored in a Charged Capacitor Potential energy Energy density Charging a capacitor by transferring charge through the space between the plates Energy in a capacitor can be considered to be stored in the electric field • q is the charge on capacitor at some instant
2 U = 1 C (∆V ) =
2 • ∆V=q/C is potential difference at the same instant 1
2 ε0A
d (E d ) = (ε Ad )E
2 2 1
2 2 0 • Work necessary to transfer a charge dq is dW = ∆Vdq = q
dq
C Energy density (energy per unit volume) in any electric field is
proportional to the square of the magnitude of the electric field at a • The total work required to charge the capacitor to q=Q W =∫ Q 0 Potential energy stored in
a charged capacitance: q
1
dq =
C
C
U= ∫ Q 0 qdq = given point Q2
2C uE = 1 ε 0 E 2
2 Q2 1
2
= Q∆V = 1 C (∆V )
2
2C 2 303L: Capacitance and Dielectrics (Ch.26) 303L: Capacitance and Dielectrics (Ch.26) Capacitors with Dielectrics Capacitors with Dielectrics Capacitance increases dielectric constant • A parallelplate capacitor has chare Q0 and a
capacitance C0 (∆V0= Q0/C0) dielectric
constant: C = κ C0 • A dielectric is inserted between the plates
• Voltmeter indicates a decrease in the voltage κ >1 • Increase in between the plates to ∆V capacitance ∆V0 • Increase in ∆V = C= κ Q0
Q0
Q
=
= κ 0 = κ C0
∆V ∆V0 κ
∆V0 maximum
operating voltage
• Mechanical
support between
the plates Capacitance of a parallelplate
capacitor filled with dielectric 303L: Capacitance and Dielectrics (Ch.26) C =κ ε0A
d
303L: Capacitance and Dielectrics (Ch.26) 2 Capacitors with Dielectrics Electric Dipole in an Electric Field Types of capacitors • Tubular capacitor Dipole moment metallic foil interlaced with thin sheets • Electric dipole of paraffinimpregnated paper, rolled into a cylinder consists of two charges of equal magnitude q and opposite sign
• Highvoltage capacitors interwoven metallic plates separated by a distance 2a immersed in silicone oil
• Electrolytic capacitor metallic foil in electrolyte (not reversible)
• Variable capacitors p = 2aq • Electric dipole moment r r r τ = p× E
two interwoven sets of metallic • Torque on the dipole is τ = 2Fa sin θ = pE sin θ • Potential energy of the dipolefield system plates θf θf θi θi ( U f − U i = ∫ τdθ = pE ∫ sin θdθ = pE cos θ i − cos θ f
θ
θ U = − pE cos θ Polar molecules: 303L: Capacitance and Dielectrics (Ch.26) Nonpolar molecules: 303L: Capacitance and Dielectrics (Ch.26) Dielectrics SUMMARY an atomic description Capacitance and Dielectrics
• Capacitor • Electric field between the plates of a parallel between the plates:
• Electric field polarizes the dielectric • Capacitance E= E0
κ Ceq = C1 + C2 + C3 + ... • Capacitors connected in series: 1111
= + + + ...
Ceq C1 C2 C3 Q2 1
2
= Q∆V = 1 C (∆V )
2
2C 2
• Dielectric material between the plates of capacitor increases its
• Energy stored in a capacitor: induced electric field opposite to E0 capacitance: U= C = κ C0 • Electric dipole moment: E = E0 − E ind 303L: Capacitance and Dielectrics (Ch.26) Q
∆V • Capacitors connected in parallel: induced • The induced surface charge give rise to an σ ind C= • The SI unit of capacitance is farad (1F=1 C/V) surface charge is formed ⇒ two conductors carrying charges of equal magnitude and opposite sign plate capacitor without dielectric is E0
• Electric field after inserting a dielectric σ
σσ
= − ind
κε 0 ε 0 ε 0 ) rr
U = − p⋅ E p = 2aq • Torque on an electric dipole in a uniform electric field: κ −1
=
σ
κ • Potential energy of the dipolefield system:
Examples 67 r r r τ = p× E rr
U = − p⋅ E 303L: Capacitance and Dielectrics (Ch.26) 3 ...
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 Spring '09
 TSOI
 Capacitance, Energy

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