He ariā nui te rerenga hiko i roto i te ahupūngao, inā koa i roto i te ako i te aukume hiko. E whakaahua ana i te maha o ngā rārangi āpure hiko e whiti ana i roto i tētahi mata. I roto i tēnei tuhinga, ka matapakihia e mātou te tātai rerenga hiko, ngā ariā taketake kei muri, ōna tono i roto i te oranga o ia rā, me tōna whanaungatanga ki te ture a Gauss.
Ariā Taketake o te Rerenga Hiko
Ko te rerenga hiko (\(\Phi_E\)) he ine i te nui o te papa hiko (\(\mathbf{E}\)) e whiti ana i roto i tētahi mata. He rite ki te ariā o te rerenga aukume i roto i te aukume. Ko te rerenga hiko e whakawhirinaki ana ki te papa hiko, te horahanga mata e uruhia ana e te papa hiko, me te koki i waenganui i te papa hiko me te mata.
Mā te pāngarau, ka taea te whakaatu i te rerenga hiko penei:
\[ \Phi_E = \mathbf{E} \cdot \mathbf{A} \]
Kei hea:
– Ko te rerenga hiko te \(\Phi_E\).
– Ko te mara hiko ko \(\mathbf{E}\).
– Ko \(\mathbf{A}\) te ira horahanga mata.
Te Tātai Rerenga Hiko i roto i te Āhua Whakauru
Mō te take whānui ake ina kore e ōrite te papa hiko me te mata, ka taea te tatau i te rerenga hiko mā te whakamahi i te tauwehenga mata:
\[ \Phi_E = \int_S \mathbf{E} \cdot d\mathbf{A} \]
Kei hea:
– Ko te rerenga hiko te \(\Phi_E\).
– Ko te \(\mathbf{E}\) te papa hiko i ia pūwāhi o te mata.
– Ko te \(d\mathbf{A}\) he huānga whārite horahanga mata iti rawa.
E tohu ana te whārite \(d\mathbf{A}\) i te ahunga noa (poutū) o te huānga horahanga iti \(dA\) i runga i te mata \(S\).
Te Ture a Gauss
Ko te ture a Gauss tētahi o ngā whārite e whā a Maxwell e tūāpapa ana i te ariā o te aukume hiko. E kī ana tēnei ture ko te rerenga hiko katoa e puta ana i tētahi mata kua kati he rite ki te utu katoa i roto i taua mata. Mā te pāngarau, ko te ture a Gauss e whakaatuhia ana penei:
\[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{in}}}{\epsilon_0} \]
Kei hea:
– He tauwehenga mata kati a \(\oint_S\).
– Ko te mara hiko ko \(\mathbf{E}\).
– Ko te \(d\mathbf{A}\) he huānga whārite horahanga mata.
– Ko te \(Q_{\text{in}}\) te tapeke o te utu i roto i tētahi mata kua kati.
– Ko te \(\epsilon_0\) te āheinga korehau (pūmau hiko).
Mā te ture a Gauss ka taea e tātou te tatau i te mara hiko mai i tētahi tohatoha utu ōrite, he māmā ake i te whakamahi tika i te ture a Coulomb.
Ngā Whakamahinga o te Rerenga Hiko
1. Kapasitor
He taputapu te pūnga hiko hei penapena i te pūngao hiko i roto i tētahi mara hiko. Ka taea te whakamahi i te rerenga hiko i roto i te horopaki o tētahi pūnga hei whakatau i te whanaungatanga i waenga i te utu, te mara hiko, me te pūmanawa hiko. I roto i tētahi pūnga hiko whakarara, ka taea te kī he ōrite te mara hiko i waenga i ngā pereti e rua, nō reira ka taea te tatau ngāwari i te rerenga hiko.
2. Te Papa Hiko e Huri Ana i te Utu
Ka taea te whakamahi i te ture a Gauss hei tatau i te āpure hiko e karapoti ana i tētahi utu pūwāhi. Hei tauira, mō tētahi utu pūwāhi \(Q\), ka taea te tatau i te āpure hiko i te tawhiti \(r\) mai i te utu mā te whakamahi i te ture a Gauss me tētahi mata Gaussian porowhita.
3. Te Tohatoha o te Utu ki ngā Kaiārahi
I roto i tētahi kaiārahi hiko, ka tohatohahia te utu hiko kia kore ai te āpure hiko i roto i te kaiārahi hiko. Mā te whakamahi i te ture a Gauss, ka taea e tātou te whakatau i te tohatoha utu i runga i te mata o tētahi kaiārahi hiko.
4. Ngā Hiko Pūmau i roto i ngā Rauemi Whakakore Hiko
He mea nui anō hoki te rerenga hiko i roto i te rangahau i ngā rauemi dielectric, arā, ngā rauemi ka taea te whakapororaru e te papa hiko. He maha ngā whakamahinga o ngā dielectric, tae atu ki te whakamahana taura me ngā pūnga.
Tauira Tātaitanga Rerenga Hiko
Me whakaaro tātou ki ētahi tauira kia pai ake ai tō mārama ki te tatau i te rerenga hiko.
Tauira 1: Papa Hiko Kotahi
Whakaarohia he papa hiko ōrite \(\mathbf{E}\) e uru ana ki tētahi mata papatahi o te horahanga \(A\) i tētahi koki \(\theta\) ki te ahunga o te papa hiko. Ka taea te tatau i te rerenga hiko penei:
\[ \Phi_E = EA \cos \theta \]
Mena kei te poutū te papa hiko ki te mata (\(\theta = 0^\circ\)), ko te rerenga hiko ko:
\[ \Phi_E = EA \]
Tauira 2: Te Utu Pūwāhi i Roto i te Mata o te Porowhita
Whakaarohia he utu pūwāhi \(Q\) kei roto i te mata o te porowhita he radius \(r\). Ko te papa hiko i te tawhiti \(r\) mai i te utu pūwāhi ko:
\[ E = \frac{Q}{4 \pi \epsilon_0 r^2} \]
Mā te whakamahi i te ture a Gauss, ko te rerenga hiko e puta ana i te mata o te porowhita ko:
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = E \cdot 4 \pi r^2 = \frac{Q}{\epsilon_0} \]
Whakamutunga
He ariā nui te rerenga hiko i roto i te aukumetanga hiko e whakaahua ana i te nui o te āpure hiko e whiti ana i roto i tētahi mata. Mā te mārama ki te tātai rerenga hiko me te ture a Gauss, ka taea e tātou te tatau ngāwari ake i te āpure hiko mai i ngā momo tohatoha utu. Ko ngā tono o te rerenga hiko ko ngā taputapu pēnei i ngā pūnga, ngā tohatoha utu i roto i ngā kaiārahi, me ngā rauemi dielectric. Mā te mārama hohonu ki te rerenga hiko me te ture a Gauss ka taea e tātou te mārama ake me te whakamahi i ngā mātāpono taketake o te aukumetanga hiko ki ngā momo hangarau me ngā āhuatanga taiao.
Ko te tumanako ka whakaratohia e tēnei tuhinga he māramatanga taketake ki ngā kaipānui mō te ariā o te rerenga hiko, ōna tauira e pā ana, me ōna whakamahinga i roto i te oranga o ia rā me te hangarau hou. He mea nui tēnei māramatanga ehara i te mea i roto i te ahupūngao anake engari i roto hoki i ngā momo kaupapa hangarau me te pūtaiao.
