1 1. Conductors A conductor is a material in which the electrons at the outer periphery of an atom have no great affinity for any particular individual atom; they are not bounded to individual atom. The electrons can move freely in the conductors. EeF rr ?= E r e § 18.1 Conductors in electric field 2 2. The conditions of electrostatic equilibrium of a conductor 1Charge separation continues until such time that the total electric field within the conductor is zero. Conductor is a equipotential body. 2The free charge Q on a conductor does not lie within it but must reside on its surface. VE ??= r § 18.1 Conductors in electric field 3. The magnitude of the electric field at the surface of a conductor 1In electrostatics, the electric field vector at the surface of a conductor must be perpendicular to the surface. The surface of the conductor is an equipotential surface. 2The magnitude of the electric field at any location on the surface of any conductor is equal to to the magnitude of the local surface charge density σ divided by ε 0 . 0 ε σ =E ∫ ?=? path any drEV r r ? § 18.1 Conductors in electric field 3 0 0 SSS 0 S d d d d d 1 d rightleftletera ε σ ε σ ε = ? =?=?+?+?=? =? ∫∫∫∫ ∫∫ ?? E S SESESESESE QSE S Q rrrrrrrr rr S r ? S r ? § 18.1 Conductors in electric field 4.Point discharge—lightning rod E r E r E r σ∝E For differently shaped conductors , or for all shapes of conductors in the presence of other conductors or point charges the surface charge density varies with position on the conductor. § 18.1 Conductors in electric field 4 r R RQ rq E E R r Q q R Q E r q E R Q r q V R r Rr === == == 2 2 2 0 2 0 00 / / ; 44 44 πεπε πεπε R r Q q For a conductor of arbitrary shape, the electric field at its surface has the greatest magnitude near those portions with the smallest radius of curvature. § 18.1 Conductors in electric field 5. An isolated conductor with cavity If there are no charges in the cavity, then the all charges on the conductor will reside on the out surface of the conductor. The electric field inside the cavity is zero. 0 0d 1 d S 0 in = ==? ∫∫ σ σ ε SSE S rr § 18.1 Conductors in electric field 5 s + ? Is this situation possible? No, in this case, the conductor is not a equipotential body. ? ? ? ? + + + + + + + + + + + 0=′+ EE rr 6.Electrostatic shield § 18.1 Conductors in electric field + + + + + + + + + qq ?+ ? ? ? ? ? ? ? q + + + + + No charge on the conductor cavity + + + + + + + + + qq ?+ ? ? ? ? ? ? ? qQ+ + + + + + + + + + + A charged conductor cavity § 18.1 Conductors in electric field 6 + + + + + + + + + qq ?+ ? ? ? ? ? ? ? q + + + + + + + + + + + + + + qq ?+ ? ? ? ? ? ? ? § 18.1 Conductors in electric field Van De Graaff electrostatic generator § 18.1 Conductors in electric field 7 Exercise 1: Q a , Q b and S is known, find the σ 1 ,σ 2, σ 3 , and σ 4 . + 4321 σσσσ a Q a b S b Q ? ? 1 P 2 P conductor )2( )1( 43 21 b a QSS QSS =+ =+ σσ σσ Charge conservation )3(0 2222 0 4 0 3 0 2 0 1 1 =???= ε σ ε σ ε σ ε σ P E )4(0 2222 0 4 0 3 0 2 0 1 2 =?++= ε σ ε σ ε σ ε σ P E The electrostatic equilibrium § 18.1 Conductors in electric field S QQ S QQ ba ba 2 2 32 41 ? =?= + == σσ σσ + 4321 σσσσ a Q a b S b Q ? ? 1 P 2 P conductorThe results are If baba QQQQ =<> ,0,0 σσσ σσ ==?= == S Q a 32 41 0 There are no charges on the outboard surface of the conductors. § 18.1 Conductors in electric field 8 § 18.2 The dielectric in the electric field 1. Dielectric There is no freely mobile electron in dielectric, all the electrons are bounded in individual atoms. 2. Dielectric in an electric field 1polar dielectrics Ep r rr ×=τ 2nonpolar dielectrics EqF rr = ??? ? ? 0 E r + + + + + - - - - - + E′ r H H H H C + - - + - + - + - + - + - + - + - + - + - + - The result is emerge the surface charge on the both sides of the dielectric slab. EEE ′ += rrr 0 3. The electric field in the dielectric § 18.2 The dielectric in the electric field 9 For instance: put a dielectric slab in the two conductor planes with charge +Q and –Q, respectively. 0 EE < The electric field in the dielectric + + + + + + + ? ? ? ? ? ? ? + + + + + ? ? ? ? ? 0 E r E′ r E r E′ r Q+ Q? We can prove that 0 0 0 ε σ κε σ =<= EE 0 κεε = κ is called dielectric constant. is called permittivity of dielectric. § 18.2 The dielectric in the electric field § 18.3 Capacitors and Capacitance 1. Capacitor A capacitor is a device of storing the charges and the electric energy. Two conductors, isolated electrically from each other and from their surroundings, form a capacitor. 10 When the capacitor is charged, the conductors, or plates have equal but opposite charges of magnitude Q. For an isolated , ideal, charged capacitor, the charge separation can last indefinitely § 18.3 Capacitors and Capacitance 2. Capacitance The capacitance of a capacitor is defined to be the ratio of the absolute value of the charge on either conductor to the absolute value of the potential difference between them: V Q C = (C/V)[farad(F)] The capacitance of a capacitor is a measure of its capacity for holding (storing )charge. Note: The value of the ratio is independent of either Q or V. § 18.3 Capacitors and Capacitance 11 3. Calculating the capacitance 1Calculating the electric field; 2Calculating the potential difference; 3Assuming the absolute charge is Q on one of the plates; 4Using the definition of the capacitance. Example 1: A parallel—plate capacitor EAQ QSE i i S 0 0 1 d ε ε = =? ∑ ∫ rr Q Q § 18.3 Capacitors and Capacitance EdrErEV rEVVV d ==?= ??=?=? ∫∫ ∫ ? + ? + +? 0 dd d0 r r r r d A d A Ed Q V Q C 0 0 )/( ε εσ σ ==== Q Q § 18.3 Capacitors and Capacitance 12 Example 2: A cylindrical capacitor a b L rL Q E rLEQ QSE i i S 0 0 0 2 )2( 1 d πε πε ε = = =? ∑ ∫ rr r § 18.3 Capacitors and Capacitance a b L Q r r L Q rEV rEVVV b a ln 2 d 2 d d0 00 πεπε ==?= ??=?=? ∫∫ ∫ ? + ? + +? r r r r a b L a b L Q Q V Q C ln 2 ln 2 0 0 πε πε === § 18.3 Capacitors and Capacitance 13 Example 3: A spherical capacitor 2 0 )(inside 0 4 1 d r Q EQSE S i S πεε ==? ∑ ∫ rr )( 4 21 2 0 RrR r Q E <<= πε ) 11 ( 4 d 4 d 210 2 0 2 1 RR Q r r Q rEV R R ?==?= ∫∫ ? + πεπε r r 12 21 0 4 RR RR V Q C ? == πε R 2 1 R § 18.3 Capacitors and Capacitance Example 4: an isolated sphere The second conductor is considered to be at infinity. R r Q R Q V 0 4πε = R V Q C 0 4πε== 4. Capacitor with a dielectric κκε σ 0 0 E E ==The electric field § 18.3 Capacitors and Capacitance 14 κ 0 V V = 0 0 C V Q V Q C κκ === d A d A CC εε κκ === 0 0 )(ln 2 )(ln 2 0 0 ab L ab L CC πεπε κκ === 12 21 12 21 00 44 RR RR RR RR CC ? = ? == πεπεκκ § 18.3 Capacitors and Capacitance If the potential difference between the plates of a capacitor is maintained, the effect of a dielectric is to increase the charge on the plates. § 18.3 Capacitors and Capacitance 15 If the charge on the capacitor plates is maintained, the effect of a dielectric is to reduce the potential difference between the plates. § 18.3 Capacitors and Capacitance 5. Series and parallel combinations of capacitors 1capacitors in parallel VCqVCqVCq 332211 ,, === VCqqqq eq =++= 321 321 CCCC eq ++= § 18.3 Capacitors and Capacitance 16 2capacitors in series 3 3 2 2 1 1 ,, C q V C q V C q V === ) 111 ( 321 321 CCC qVVVV ++=++= 321eq 1111 CCCq V C ++== § 18.3 Capacitors and Capacitance Exercise 1: A parallel-plate capacitor of plate area A is filled with two dielectrics as in Fig. (a) What is the capacitance of the capacitor? If the two dielectrics are filled as in Fig. (b) What is the capacitance of the capacitor? § 18.3 Capacitors and Capacitance 17 Solution: The capacitor can be viewed as two capacitors C 1 and C 2 in parallel, ) 2 ( )2/()2/( 210 2010 21 κκε κεκε + = += += d A d A d A CCC § 18.3 Capacitors and Capacitance The capacitor can be viewed as two capacitors C 1 and C 2 in series, Solution: )( 2 2/2/ 2/2/ 21 210 2010 2010 21 21 κκ κκε κεκε κεκε + = + = + = d A d A d A d A d A CC CC C § 18.3 Capacitors and Capacitance 18 Exercise 2: What is the capacitance of the capacitor, of plate area A, shown in Figure? Solution: ) 2 ( 4 )2/()2/( )2/)(2/)(/( 4 32 32 1 0 32 3200 32 32 1 κκ κκ κ ε κκ κκεε + += + += + += d A dA d A CC CC CC § 18.3 Capacitors and Capacitance § 18.4 The energy stored in a capacitor The charge on the plates: The potential between the plates: V Q → → 0 0 1. The work done by external force 0 0 Q? V Q qd q+ q? u E r elec F r + + exelec FF rr ?= 0)(dddd exelectotal ==+= KEWWW 19 C Q q C q WWPE Q 2 dd 2 0 exex ==== ∫∫ q C q quPEWW dd)0()(ddd elecex =??==?= QVCV C Q W 2 1 2 1 2 2 2 === 2. Energy stored in an electric field EdV = d S C 0 ε = Foe a parallel-plate capacitor § 18.4 The energy stored in a capacitor 2 0 2 1 E V W w e ε== VEdE d S CVW 2 0 22 0 2 2 1 2 1 2 1 ε ε === 3. The energy density VEVEVwW VVV e d 2 1 d 2 1 d 2 0 2 0 εε ∫∫∫ === The energy stored in the electric field § 18.4 The energy stored in a capacitor