Te whakatau i te mara hiko mā te whakamahi i te ture a Gauss

Te papa hiko nā te utu kotahi

Hei tatau mara hiko i hangaia e te utu pai kotahi, ko te taahiraa tuatahi ko te whiriwhiri i tētahi mata Gaussian porowhita he radius r kei reira te pokapū o te porowhita i te utu kotahi. Ko te horahanga o te mata o te porowhita he 4πr².

Ka uru poutū te papa hiko e puta mai ana i waenganui o te pōro ki te mata o te pōro kia puta ai te tātai rerenga hiko ko Φ = E A. Tātai Te ture a Gauss ko Φ = Q / ε_o

Ko te papa hiko i tētahi pūwāhi i te tawhiti r mai i tētahi utunga kotahi ko:

Whakaahuatanga: E = te papa hiko, k = te pūmau o Coulomb (9 x 10⁹ Nm²/C²), Q = te utu hiko, r = te tawhiti mai i te utu hiko

Koinei te tātai mō te āpure hiko i whakaputaina e te utu hiko. Ka taea te whakaputa i tēnei tātai mā te whakamahi i Te ture a Coulomb.

Ngā papa hiko i roto, i waho hoki i tētahi porowhita totoka ōrite e utua ana te hiko

He porowhita totoka he rite te utu hiko, he ōrite rānei, ko te utu katoa he Q, ko te rōrahi V = 4/3. π R³ ā, ko te kiato utu i roto i tētahi porowhita totoka ko ρ = Q/V. Whakatauhia te kaha hiko te papa hiko i roto i te pōro, i waho hoki i te pōro.

a) Papa hiko i roto i tētahi pōro totoka

Ko te pūtoro o te porowhita totoka he R, ko te mata Gaussian ia i whiriwhiria he porowhita he pūtoro r, ko r < R. Ko te rōrahi o te porowhita totoka ko V, ā, ko te rōrahi o te porowhita Gaussian ko V'.

Ko te utu hiko i roto i te porowhita Gauss ko:

Ka uru poutū te papa hiko e puta mai ana i waenganui o te pōro ki te mata o te pōro, kia rite ai te tātai mō te rerenga hiko Φ = E A. Ko te tātai ture a Gauss ko Φ = Q / ε_o

Ko te papa hiko i tētahi pūwāhi i te tawhiti r mai i te pokapū o te porowhita totoka ko:

I runga i te tātai i runga ake nei, he rite te āpure hiko (E) ki te utu hiko (Q) me te pūtoro o te mata Gaussian (r), he rite whakamuri ki te pūtoru o te pūtoro o te porowhita totoka (R).³).

b) Papa hiko i waho o te porowhita totoka

Ko te pūtoro o te porowhita totoka he R, ko te pūtoro ia o te mata Gaussian porowhita he r, ko te r > R. Ko te utu o te porowhita totoka he Q; kei roto te porowhita totoka i te porowhita Gaussian, nō reira ko te utu hiko i roto i te porowhita Gaussian he Q.

Ka puta te papa hiko mai i waenganui o te pōro, ka uru poutū ki te mata o te pōro, nō reira ko te tātai mō te rerenga hiko ko Φ = E A. Ko te tātai ture a Gauss ko Φ = Q / ε_o

Ko te papa hiko i te tawhiti r mai i te pokapū o te porowhita totoka ko:

Ko te papa hiko i roto, i waho hoki i te anga o tētahi porowhita tuwhera kua rite te utu hiko

He porowhita tuwhera me te pūtoro R me te rōrahi V = 4/3 π R³, he rite tonu te utu hiko pai i runga i tōna kiri me te utu katoa QWhakatauhia te kaha o te papa hiko i roto, i waho hoki o te anga porowhita.

a) Te papa hiko i roto i te porowhita kau

Nō reira, ko te āhua o te porowhita tuwhera he hiko kei runga i te mata o te porowhita, engari kāore he hiko kei roto i te porowhita. Mena he porowhita te mata Gaussian kua whiriwhiria, ā, kei roto te porowhita Gaussian i tētahi porowhita tuwhera, kāti kāore he hiko kei roto i te porowhita Gaussian. Kore te hiko, nō reira kore anō hoki te papa hiko. Nō reira, kore te papa hiko i roto i te porowhita tuwhera.

b) Te papa hiko i waho o te porowhita kau

Ko te porowhita tuwhera he R te pūtoro, ko te mata Gaussian ia i whiriwhiria he porowhita me te pūtoro r, ko r > R.

Ka uru poutū te papa hiko e puta mai ana i waenganui o te pōro ki te mata o te pōro, kia rite ai te tātai mō te rerenga hiko Φ = EA = E 4π r²Ko te tātai mō te ture a Gauss ko Φ = Q / ε_o .

Ko te papa hiko i tētahi pūwāhi i te tawhiti r mai i te pokapū o te porowhita tuwhera ko:

Papa hiko e tata ana ki tētahi waea angiangi kua utua he hiko

He waea angiangi, he mutunga kore te roa, he rite tonu te utu hiko pai, ā, he matotoru te utu o λKo te utu hiko i runga i te waea ko Q = λl. Tātaihia te kaha hiko. te papa hiko e karapoti ana i te waea angiangi.

Kua whiriwhiria te mata Gaussian kia porotaka te āhua, ko te roa ko te l, ko te radius ko te r. E rua ngā momo mata, arā, he mata porowhita kei ngā pito e rua o te rango te radius ko te r (ko te horahanga o te mata ko πr²) me te mata porotaka me te roa l (ko tōna horahanga mata he 2πr l).

He pai te utu hiko, nō reira ka puta te papa hiko i te waea e poutū ana ki te mata o te ngongo, kia whai uara ai te rerenga hiko Φ = EA = E 2πr l. I tētahi atu taha, he whakarara te papa hiko ki ngā pito e rua o te ngongo porowhita, nō reira he kore te rerenga hiko.

Ko te papa hiko i tētahi pūwāhi i te tawhiti r mai i te waea ko: