Chapter 26 Capacitance and
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1 hapter 6 apacitance and Dielectrics We must do work, q, to bring a point charge q from far away ( at infinity) to a region where other charges are present. The work done is stored in electrostatic field energy. 6. Definition of apacitance 6. alculating apacitance Measure of the capacity to store charge: Unit: farad (F): F / ; µf 6 F; pf F The ratio of charge to the potential depends on the size and the shape of the conductor. apacitors A device consisting of two conductors carrying equal but opposite charges is called a capacitor. Parallel Plate apacitor E A, A d E, Ed d A A
2 ylindrical apacitor πrle > E πrl a b a b dr ln πrl L a π b π L b ln a Spherical apacitor 4π a r E, dr 4π r 4 a b b π ab 4π b a Selfapacitance The potential of a spherical conductor of radius carrying a charge is The selfcapacitance of a spherical conductor is: k 4π k k. 6. ombination of apacitors apacitors onnected in Parallel Obtain and to calculate. & /
3 ( ) / apacitors connected in parallel: eq 4... apacitors connected in series & / series > sum voltage & parallel > sum charges apacitors connected in series: 4... Example: Two capacitors are removed from the battery and carefully connected from each other. & & 96µ apacitors in Series and in Parallel??,, & > &
4 > 6.4 Energy Stored in a harged apacitor du dq q dq U du q dq Electrostatic Field Energy (derived from energy stored in a capacitor) A, Ed d U A d Electrostatic Energy Density: ( Ed ) E Ad E U u e E (energy per unit volume) Example: alculate the energy stored in the conductor carrying a charge. r < : E k r > : E r du u d e k ( 4πr dr) ( πr dr) E 4 4 r U du π k r k dr 4
5 6.5 apacitors and Dielectrics When the space between the two conductors of a capacitor is occupied by a dielectric, the capacitance is increased by a factor κ ( κ > ) that is characteristic of the dielectric. If the dielectric field is E before the dielectric slab is inserted, after the dielectric slab is inserted between the plates the field is E E > the potential is κ If Ed Ed κ κ ' κ., the capacitor is / κ The capacitance of a parallelplate capacitor filled with a dielectric of constant κ is Aσ Aσ κ κ E d Aσ κ σ d / permittivity of the dielectric. Aκ A d d acuum: κ v > κ is called the material: κ m > the Dielectric: κ > 5
6 Energy Stored in The Presence of a Dielectric The energy stored in a capacitor is: q q du dq > du dq > du U dq > U The energy of a capacitor with the dielectric is U u e E A d κe A d ( Ed ) ( Ed ) E ( Ad ) material: κ m > acuum: κ v E E the Dielectric: κ >. You lose electric force to separate the charge.. You enlarge the charging capacity as you know the dielectric will breakdown in a high electric field. (If the same charge > you lose some electric field,). You increase the energy per unit volume. ombination of apacitors Example: A parallelplate capacitor has square plates of edge length cm and a separation of d 4 mm. A dielectric slab of constant κ has dimensions cm X cm X 4 mm. (a) What is the capacitance without the dielectric? (b) What is the capacitance with the dielectric? (c) What is the capacitance if a dielectric slab with dimensions cm X cm X mm is inserted into the 4mm gap? (a) A, (b) d (c) series connection: A d A A d / 4 d / 4 A Aκ d / 4 d / 4 Example: The parallel plates of a given capacitor are square with A a and separation distance d. If the plates are maintained at a constant potential and a 6
7 square of dielectric slab of constant κ, area A a, thickness d is inserted between the capacitor plates to a distance x as shown in the following figure. Let σ be the free charge density at the conductorair surface. (a) alculate the free charge density σ κ at the capacitordielectric surface. (b) What is the effective capacitance? (c) What is the magnitude of the required force to prevent the dielectric slab from sliding into the plates? (a) In air: (b) (c) σ σ K Kσ E & d σ, in dielectric: ( a x) axk a a d d d U σ σ K E & d K K K ( a ( K ) x) d d d d a F K dx dx dx dx d ( ) The first term is due to charge redistribution and the second is due to the additional charges supplied by the constant voltage. Example: A parallel plate capacitor with plates of area LW and separation t has the region between its plates filled with wedges of two dielectric materials. Assume t is much less than both W and L. (a) Please determine its capacitance. The thickness of the k material decrease as a function of k material is of tx L For a short stripe of dx, The series connected capacitance A Wdx d t The total parallel connected capacitance ( κ ) ( L x) / L,, t ( L x) L ( κ ) A Wdx d tx / L k L W dx t κ κ κ while that of the x L k W total L W dx W L κ ln x κ t t κ κ κ L κ κ 7
8 6.6 Electric Dipole in an Electric Field Inside the material to make sure that the electric field lines are from the positive charge to the negative charge If the field is uniform, it can rotate the dipole. r r r r r r r r r τ qe qe qd E p v r r τ p E pe sinθ ( ) E The potential energy: du Fdx τdθ ( pe sinθ ) dθ θf U pe sinθdθ pe( cosθ cosθ ) θi f i q q E U r r p E 6.7 An Atomic Description of Dielectrics 8
9 bound charge total < total > total E bound charge E Example: A hydrogen atom consists of a proton nucleus of charge e and an electron of charge e. The charge distribution of the atom is spherically symmetric, so the atom is nonpolar. onsider a model in which the hydrogen atom consists of a positive charge e at the center of a uniformly charged spherical cloud of radius and total charge e. Show that when such an atom is placed in a uniform external field E r, the induced dipole moment is proportional to E r r r ; that is, p αe, where α is called 9
10 the polarizability. E 4π e r 4π A 4πr e 4π inside _ from _ e er p er α E α > 4π α 4π r Magnitude of The Bound harge b σ b E & E E E κ E E b σ f > E b E κ > σ b σ f κ σ σ σ σ σ κ σ κσ effective f b f f
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