v U = e i,..., e n ) v = n v i e i i= e i i v T v = = v v n v n U v v v +q 3q +q +q b c d XY X +q Y 3q r +q = r 3q = r +q = r +q = r 3q = r +q =
E = E +q + E 3q + E +q = k q r+q 3 + k 3q r 3q 3 b V = kq ) + k q r+q 3 = kq 4 3 ) 4 + c +q +q 3q W = kq 3kq ) + = kq 6 ) + c r = x y T r x +q = y r 3q = x + y r x + +q = y V x, y) = kq x + y ) 3 x + ) + y + x + ) + y ) x Ex, y) = V x, y) = kq x + y ) ) 3x + ) 3 x + ) + y ) + x + 3 x + ) + y ) ) 3 y ) x + y ) ) 3y 3 x + ) + y ) + y 3 x + ) + y ) ) 3 x = y = q d b c d d e d
X r = x y T d ± = d d ± + r ± = r x r ± = r d ± = d = x ± d y y E = E + + E = k q r+ 3 r + + k q r 3 r ± = b y = x = c r+ 3 r = kq r 3 + x ± d ) + y E y = ) = kq x + d ) E x = ) = kq d r 3 d 4 + y x + d y x d ) 3/ e x ) r 3 e x x d y d/ r 3 ± = x ± d ) 3 = ) 3 3 x + y + y ± d ) x x + y ) 5 +... r 3 3 xd r 5 d E r d) = kq r 3 3 xd r 5 x y ) µ = qd + qd = q d d = qd e x
E r d) = k µ µr + 3k r3 r 5 r d V = kq ) r + r r + ) = x r ± r± 3 x ± d ) r ) y r ± = r± 3 y e r+ r d/ r ± r xd r 3 V r d) = kq xd r 3 = k µr r 3 r = x y T µ = µ x µ x T r = x + y r d x V µ x E = k r 3 3µ xx + µ y y r 5 x µ y r 3 3µ xx + µ y y = k r 3 r 5 y d V x, y) = k µ xx + µ y y x + y ) 3/ V x = k µ x r 3 3k µ xx + µ y y r 5 x µ x µ y + 3k µ xx + µ y y r 5 x y = k µ µr + 3k r3 r 5 r Q µ b c A α A A A
V µ r) = k µr r 3 W = QV µ r) = kq µr r 3 b µr = W = c r Q α π α W = Q V r ) V r) = kq µr r 3 cosπ α) = cos α kqµr cosπ α) µr cos α = kqµr r3 r 3 kq r 3 W = kq µ cos α r +3q q µ 9 b c Y q X +3q cos45 ) µ = µ sin45 = µ ) E = k q 3 + 3k q3 k µ ) 3 = k 3q µ 4 q µ 4
b V = k q + 3k q = k q c q W q = qv µ r) = kq µr qµ π ) = k 3 cos 4 +3q q W +3q = 3q V µ r ) + V q r ) ) = 3q k µr 3 k q ) = 3q k µ 3π cos 4 ) k q cosπ/4) = / cos3π/4) = sinπ/4) = / W = W q + W +3q = 4k qµ 3k q < m l d Q x Q + Q α d x r ± = d ± x x F ± = k Q ± r ± F g = mg T mg = T cos α T = mg cos α k Q ± r ± = T sin α = mg sin α = mg tn α cos α
tn α = x l 4 x Q ± k d ± x) = mgx l 4 x Q ± = d ± x) mgx k l 4 x Q + > Q d = Q + = mgx 3 k l 4 x +q +q 3q +q b c 3q 4q r +q = r +q = µ = q 3 3 / / r 3q = 3 r +q = 3 r +q = 3 r 3q = 3 6 6 3
c b µ = q 3 µ = q 3 3 +q q +q +q q d b c X r + = e x r = ) + d e x r + = e x r = ) d e x Q µν = k q k 3Rkµ R kν Rk δ ) µν Q µν µ ν Y Z Q xx Q xx = k q k R k Q yy Q zz Q xx Q xx + Q yy + Q zz = Q xx + Q yy = Q yy = Q zz = Q xx ) ) ) ) Q xx = q + d + q + q q + d =
= q q ) ) ) q + d = q ) + d ) ) ) + d = ) + + d = qd + d) Q yy = Q zz = qd + d) = qd + d) b V = k r 5 Q µν x µ x ν = k Qxx r 5 x + Q yy y + Q zz z ) µ ν Q xx x Q xxy Q xxz = Q xx x y + z ) α X β Y Y Z x = r cos α y = r sin α cos β z = r sin α sin β x y + z ) = r 3 cos α ) V = k r 3 Q xx 3 cos α ) = k r 3 qd + d) 3 cos α ) β c E = V = k x r 5 Q xx y = k cos α qd + d) r4 sin α cos β z sin α sin β β +q b b c ϱ σ + < b < c c
λ σ S r E ds E ds = E ds S E ds = E S ds = 4πr E = q ϵ E = 4πϵ q r b Q = 4 3 πb3 3 )ϱ Q + = 4πc σ + r r S E ds = E I = r b b r < c r > c r = c S E ds = 4πr E = 4πr3 3 )ϱ E II = ϱ r 3 3 ) 3ϵ 3ϵ r 4πr E = Q 4πr E = Q + Q + = 4 ϵ ϵ 3 πb3 3 )ϱ + 4πc σ + ϵ = 4πb3 3 )ϱ E III = ϱ b 3 3 ) 3ϵ 3ϵ r E IV = ϵ r 3 b3 3 )ϱ + c σ + r V r) = r E dr = c V II r) = c V III r) = Q + Q b + 4πϵ r dr c r V IV r) = V I r) = 4πϵ Q + Q r + 4πϵ r dr c Q r 4πϵ r dr b Q r + Q + c E dr Q + Q + 4πϵ r dr = Q + Q + 4πϵ r Q 4πϵ r dr = Q 4πϵ r + Q ) + c ϱ b 3 3 ) 3ϵ r dr = Q 4πϵ r + Q ) + ϱ r c 3ϵ b + 3 r ) b ) ϱ 3ϵ b + 3 ) b c l Q = λl Q = πlσ r S
S S r < S E ds = E ds + S S E ds = E r > S ds = πrle = E II = λ + πσ πrϵ Q ϵ E I = λ πrϵ R > r < r λ + πσ V II = E dr = ln πϵ R V I = λ πrϵ ln R r ) + σ ϵ ln V x, y) = y + x A = y + x y = A x E = x A E dr E dr = dx dx dr = = dy x dx x R r R ) ) dx = x dx E dr E E +q m v b v k d
r = T v = v T F = qe = qe T m x x X x = v x = ẍ vt) = rt) = = qe qe m t v qe m t v t xy) = qe mv y b qed W = qed = m v k v ) v k = v + qe m d Y E φ I ˆαˆαˆα = digα, α, α ) ) b c E U E cos φ E = E sin φ U α µ = ˆαˆαˆα E = α α U E cos φ E sin φ U = α E cos φ α E sin φ U
b τ = µ E = e x e y e z α E cos φ α E sin φ E cos φ E sin φ = E α α ) sin φ cos φ e z I I d φ dt = E α α ) sin φ cos φ sin φ cos φ = sinφ) φ = φ d φ dt + E α α ) φ = I ω = E α α φ) = φ φ) = c I de p = µ de = α cos φ + α sin φ)e de E p = α cos φ + α sin φ) E E de = α cos φ + α sin φ)e α µ µ
r µ µ = r µ α = r µ α = µ µ E p µ, µ ) = µ k µ rµ 3 + 3 µ ) r µ µ µ µ µ rµ 5 r µ µ = k µ rµ 3 3 µ ) r µ µ )µ r µ µ ) µ rµ 5 = 3k µ 8 3 µ µ α E µ +µ E p µ, α) + E p µ, α) = αe µ +µ E µ +µ = k µ rµ 3 α + 3 µ ) r µ α rµ 5 r µ α + k µ α rµ 3 α + 3 µ ) r µ α rµ 5 r µ α = k α E µ +µ = E µ +µ E µ +µ = 4k 6 µ + µ ) E p = 3k 8 3 µ µ k 6 αµ + µ ) µ µ 3 µ µ + µ ) d ϱ Q V b d V Q c d L V Q Q ϱ + ϱ d S b ϵ c b d L b ϵ
S d U ϵ A B b c R U C b ϵ c b d c e d R I b R XY R XZ I b
I b c < b < c d ϱ v X Y b b I