ch26 - Capacitance and Dielectrics Chapter Outline...

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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 Parallel-plate 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 parallel-plate 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 parallel-plate 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 parallel-plate 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 paraffin-impregnated paper, rolled into a cylinder consists of two charges of equal magnitude q and opposite sign • High-voltage 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 dipole-field 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 dipole-field system: Examples 6-7 r r r τ = p× E rr U = − p⋅ E 303L: Capacitance and Dielectrics (Ch.26) 3 ...
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