Ngā Tauira Pātai me te Kōrero mō te Ture a Gauss
Ko te Ture a Gauss tētahi pou matua o te aukume hiko. He huarahi whai hua tēnei hei tatau i te āpure hiko e puta mai ana i te tohatoha o te utu hiko. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru me ngā whakamahinga o te Ture a Gauss i roto i ngā horopaki rerekē.
Te Ariā Taketake o te Ture a Gauss
I mua i te tīmatanga ki ngā tauira raruraru, me arotake tātou i te ariā taketake o te Ture a Gauss. E kī ana te Ture a Gauss ko te rerenga hiko katoa \( \Phi_E \) e puta mai ana i tētahi mata kua kati he rite ki te utu katoa \( q_{in} \) e karapotia ana e te mata. Mā te pāngarau, ko te Ture a Gauss e whakaatuhia ana penei:
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \]
kāore i te mana:
– Ko te rerenga hiko te \( \Phi_E \).
– Ko te mara hiko te \( \mathbf{E} \).
– Ko te ira horahanga mata te \( \mathbf{A} \).
– Ko te utu i roto i te mata kua kati ko \( q_{in} \).
– Ko te \( \epsilon_0 \) te āheinga korehau (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \)).
Tauira Pātai 1: Te Papa Hiko i roto i te Porowhita Arataki Pokapū
Pātai:
He porowhita arataki tuwhera tōu, he pūtoro o waho \( R \) me te utu katoa \( Q \). Tātaihia te āpure hiko i roto i te arataki tuwhera.
Kōrero:
– Te Whakatau i te Mata Gaussian:
Me kī ka whiriwhiria e tātou he mata Gauss porowhita porowhita me te radius \( r \) i roto i te kōhao kawe hiko (kei reira \( r < R \)). - Tātaitanga Rere me te Utu: Nā te mea he kōhao kau te roto o te porowhita kawe hiko, ko te utu i roto i te mata Gauss he kore (\( q_{in} = 0 \)). - Te Whakamahinga o te Ture a Gauss: E ai ki te Ture a Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \]
Kōrero:
Mō \( r < R \): - Te Whakatau i te Mata Gaussian: Kōwhiria he mata Gaussian porowhita he radius \( r \) kei roto i te porowhita totoka. - Te Tātaitanga o te Utu: Nā te mea he ōrite te tohatoha o te utu, ko te utu i roto i te radius \( r \) ko: \[ q_{in} = \rho \cdot \frac{4}{3}\pi r^3 \] kei reira \( \rho = \frac{Q}{\frac{4}{3}\pi R^3} \). \[ q_{in} = \frac{Q}{\frac{4}{3}\pi R^3} \cdot \frac{4}{3}\pi r^3 = Q \left(\frac{r^3}{R^3}\right) \] - Te Whakamahinga o te Ture a Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \] Nō reira: \[ E \cdot 4\pi r^2 = \frac{Q \left(\frac{r^3}{R^3}\right)}{\epsilon_0} \] Mā te whakangawari: \[ E = \frac{Q r}{4\pi \epsilon_0 R^3} \] Nō reira, ko te papa hiko i roto i te porowhita (\( r < R \)) ko: \[ E = \frac{Q r}{4\pi \epsilon_0 R^3} \] Mō \( r > R \):
– Te Whakatau i te Mata Gaussian:
Kōwhiria he mata Gaussian porowhita me te radius \( r \) kei waho o te porowhita totoka.
– Tātaitanga Utaina:
Ko te tapeke o te utu i roto i te mata Gaussian ko te tapeke o te utu o te porowhita \( Q \).
– Te Whakamahinga o te Ture a Gauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Nō reira:
\[
E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0}
\]
Mā te whakangawari:
\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]
Nō reira, ko te papa hiko i waho o te porowhita (\( r > R \)) ko:
\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]
Whakamutunga
He taputapu kaha te ture a Gauss mō te tātari i ngā mara hiko i roto i ngā momo āhuatanga. Mā te whiriwhiri i tētahi mata Gaussian e tika ana, me te whakamahi i ōna mātāpono taketake, ka taea e tātou te tatau i ngā tohatoha mara hiko kia pai ake te whai hua. Mā roto i ngā tauira i runga ake nei, kua kite tātou i ngā whakamahinga o te ture a Gauss i roto i ngā āhuatanga pēnei i te mara hiko i roto i tētahi porowhita kawe hiko, he pereti whakarewa mutunga kore, he utu pūwāhi, me tētahi porowhita kei roto he utu ōrite. Mā te mārama me te mahi tonu ka whakarite i tētahi whakamahinga pakari o te ture a Gauss i roto i ngā tono hiko aukumetanga.
