Chapter 2. Coulomb s Law and Electric Field Intensity

Transcription

1 Chapter. Culmb s Law and lectric Field Intensit Hat; 9/9/009; -1.1 The perimental Law f Culmb Frm the eperiment the frce between tw charged bjects is QQ F k : Frce in Newtn (N) where Q1 and Q : Charges in culmb(c) : eparatin between charges in meters (m) k 1 πε : Prprtinalit cnstant 4 1 ε F m 36π Culmb s law ( ) / : Permittivit f free space F QQ Culmb s law in a vectr frm The frce eerted n Q b Q 1 QQ F 1 πε 4 1 where the unit vectr 1 r r1 1 r r 1 1 The Culmb frce is mutual QQ QQ F F The Culmb s law is linear Increased charge nq1 Increased Culmb frce b the factr n everal charges act independentl The ttal frce is a sum f individual frces.

2 Hat; 9/9/009; -. lectric Field Intensit A fied charge Q 1 and a test charge Q t mving in space Q t feels different frces at different lcatins. Frce field. lectric field intensit is defined as a vectr field, Ft Q1 1t Qt 1 t It is a vectr frce n a unit charge in the unit f vlts per meter [v/m] In general Q a with directed frm the surce pint t the field pint. (9) A charge Q 1 at the rigin f the spherical crdinate sstem. The electric field intensit, Q1 a r r The inverse-square-law applies : a r is the radial unit vectr In Cartesian crdinate sstem ( ) Q + + ( + + ) + + / + + r ( ) 1 a a a 3/ Ver cmplicated since the spherical smmetr is ignred

3 Hat; 9/9/009; -3 The general frm f r is the surce pint and r is the field pint N spherical smmetr Use Cartesian crdinate sstem Use r r in (9) Q Q r r Q ( ) + ( ) + ( ) ( r ) 3/ 4 πε 4 r r r r πε r r 4 πε ( ) + ( ) + ( ) Culmb s law is linear The electric field intensit frm tw pint charges, Q1 at r 1 and Q at r Q1 Q ( r ) 1 + a r r r r 1 The electric field intensit frm N pint charges N Qm ( r ) m m 1 r r m

4 Hat; 9/9/009; -4.3 Field Due t a Cntinuus Vlume Charge Distributin Ver large number f pint charges mth cntinuus distributin Charges per unit vlume [C/m 3 ] Vlume charge densit, Fr a small charge Δ Q in a small vlume Δ v the vlume charge densit is defined as ΔQ v lim Δv 0 Δv The ttal charge within a finite vlume V is Q dv ttal V v v

5 Hat; 9/9/009; -5 The electric field intensit f a vlume charge distributin. Assume Δ Q cntained in Δ v as a pint charge. The incremental electric field intensit is ( v Δv ) r r Δ ( r ) 4 πε r r r r The ttal electric field intensit v ( r ) dv r r ( r ) V (18) 4 πε r r r r

6 Hat; 9/9/009; -6.4 Field f a Line Charge Ver large number f pint charges frm a thin line Cntinuus line charge distributin Charges per unit length [C/m] Line charge densit, L The ttal electric field intensit Ldl r r ( r ) l 4 πε r r r r First, the prper crdinate sstem shuld be selected based n the smmetr. Check if (1) ( r ) is independent f certain crdinates are er () Certain cmpnents f ( r ) Change φ and mve arund the line charge while and are fied. The line charge appears the same. is independent f φ, aimuthal smmetr. ( r ) Change and mve up and dwn alng the line charge while and φ are fied. The infinite line charge appears the same. is independent f, aial smmetr. ( r ) Change and mve radiall while and φ are fied. changes and n smmetr in this case. ( r ) Find ( r ) Therefre, the electric field intensit shuld be f the frm f ( ) at P(0,, 0) The incremental field at P Ld ( r r ) d 3 4 πε r r The ttal electric field L d 4 πε + ( ) 3/ ( ) ( + ) 3/ Ld d a a 4 πε r a : Field psitin vectr r : urce psitin vectr r r Change f variable ctθ L πε a, ignre -cmp. frm the smmetr

7 Find ( r ) at P( φ,, ) Hat; 9/9/009; -7 r + a r r r a + ( ) ( ) + a + ( ) ( ) Ldl r r d a L + a l 3/ 4 πε 4 r r r r πε + ( ) d ( ) d L + + ( ) 4 + ( ) La d ( La ) d + 3/ + ( ) + ( ) L 3/ 3/ πε 3/ L πε a (0) The same result but cmplicated calculatins b ignring the smmetr. falls ff inversel with the distance instead f squared distance. Displaced line charge The line charge is nt n -ais but displace b 6 and 8 units alng and aes. eplace the radial distance in (0) b a distance ( 6) + ( 8) ( 6) + ( 8) and â b. ( 6) + ( 8) The electric field intensit ( 6) ( 8) L + a πε ( ) ( )

8 .5 Field f a heet f Charge Ver large number f pint charges Cntinuus surface charge distributin. n a surface. Charges per unit area [C/m ] urface charge densit, The ttal electric field intensit d r r ( r ) 4 πε r r r r Hat; 9/9/009; -8 An infinite sheet f charge in -plane ( r ) shuld be independent f and crdinates. 0 due t smmetr. Divide the sheet int man narrw strips. Cnvert sheet charge t line charges d L Using (0) the differential electric field L πε d d d csθ πε + πε Integrating ver all the strips d πε + ε ( ) d at pint P ( + ) The final result an ε : a N is utward nrmal unit vectr Nte is independent f psitin. Tw parallel infinite sheets f charge A secnd sheet f charge with lcated at a. Fr > a +, ε ε The ttal electric field Fr < 0 + ε ε The ttal electric field Fr 0 < < a + ε ε The ttal electric field + + ε

9 Hat; 9/9/009; -9.6 treamlines and ketches f Fields Graphical representatin f electric fields is called streamlines, flu lines r directin lines. The tp view f electric fields frm a line charge (a) and (c) are pr representatins. (b) is OK just fr schematic. (c) is crrect methd. : N smmetr. trnger field b thicker lines. : trnger field b lnger lines. : trnger field b dense lines. Mathematical frm f streamlines In 0 plane d d ample treamlines f a line charge f L πε 1 a a + a + + The differential eq. fr the streamlines d d d ln ln + C1 d C