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 āhuatanga 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 Gaussian porowhita porowhita porowhita he radius \( r \) kei roto i te kōhao kawe (kei reira \( r < R \)).
– Te Tātaitanga o te Rerenga me te Utu:
Nā te mea he kōhao kau te roto o te porowhita kawe hiko, he kore te utu i roto i te mata Gaussian (\( q_{in} = 0 \)).
– Te Whakamahinga o te Ture a Gauss:
E ai ki te Ture a Gauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Nā te mea ko te \( q_{in} = 0 \), ko te rerenga hiko he kore anō hoki:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = 0
\]
- Whakamutunga:
Nā te mea he kore te rerenga hiko, ko te tikanga he kore anō hoki te papa hiko \( \mathbf{E} \) i ia pūwāhi i roto i te kōhao.
Nō reira, ko te āpure hiko i roto i te kōhao kawe hiko ko \( 0 \, \text{N/C} \).
Tauira 2: Te Papa Hiko e te Pereti Mutunga Kore
Pātai:
Tātaihia te papa hiko e tata ana ki tētahi pereti whakarewa mutunga kore he kiato utu mata tōna \( \sigma \).
Kōrero:
– Te Whakatau i te Mata Gaussian:
Kōwhiria he mata porotaka Gaussian "Gaussian pillbox" me ngā mata i runga ake, i raro iho hoki i te pereti, he horahanga ia mata \( A \).
– Te Tātaitanga o te Rerenga me te Utu:
Ko te rerenga hiko katoa e puta ana i ngā pito e rua o te mata ko:
\[
\Phi_E = 2EA
\]
ko \( E \) te papa hiko i ngā taha e rua o te pereti. Ko te utu katoa \( q_{in} \) e karapotia ana e te mata Gaussian ko:
\[
q_{in} = \sigma \cdot A
\]
– Te Whakamahinga o te Ture a Gauss:
E ai ki te Ture a Gauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Nō reira:
\[
2EA = \frac{\sigma A}{\epsilon_0}
\]
Mā te whakangawari:
\[
E = \frac{\sigma}{2\epsilon_0}
\]
- Whakamutunga:
Ko te papa hiko e tata ana ki tētahi pereti whakarewa mutunga kore ko:
\[
E = \frac{\sigma}{2\epsilon_0} \, \text{(N/C)}
\]
Tauira 3: Papa Hiko Huri noa i tētahi Utu Pūwāhi
Pātai:
Tātaihia te papa hiko i te tawhiti \( r \) mai i te utunga pūwāhi \( q \).
Kōrero:
– Te Whakatau i te Mata Gaussian:
Kōwhiria he mata Gaussian porowhita he radius \( r \) o te utu pūwāhi \( q \).
– Te Tātaitanga o te Rerenga me te Utu:
Ko te rerenga hiko katoa e wehe atu ana i te mata o Gaussian ko:
\[
\Phi_E = E \cdot 4\pi r^2
\]
Ko te utu katoa \( q_{in} \) e karapotia ana e te mata Gaussian ko te utu pūwāhi \( q \).
– Te Whakamahinga o te Ture a Gauss:
E ai ki te ture a Gauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Nō reira:
\[
E \cdot 4\pi r^2 = \frac{q}{\epsilon_0}
\]
Mā te whakangawari:
\[
E = \frac{q}{4\pi \epsilon_0 r^2}
\]
- Whakamutunga:
Ko te papa hiko i te tawhiti \( r \) mai i te utunga pūwāhi \( q \) ko:
\[
E = \frac{q}{4\pi \epsilon_0 r^2} \, \text{(N/C)}
\]
Tauira 4: Papa Hiko i Roto, i Waho hoki i tētahi Porowhita kei roto he Utu Kotahi
Pātai:
He porowhita totoka he radius \( R \) tōna, ā, he rite tonu te tohatoha o te utu katoa \( Q \). Tātaihia te papa hiko i tētahi pūwāhi i roto i te porowhita (\( r < R \)) me waho o te porowhita (\( r > R \)).
Kōrero:
Mō \( r < R \):
– Te Whakatau i te Mata Gaussian:
Tīpakohia he mata Gaussian porowhita me te radius \( r \) i roto i te porowhita totoka.
– Tātaitanga Utaina:
Nā te mea kua tohatohahia te utu i te taha ōrite, ko te utu i roto i te radius \( r \) ko:
\[
q_{in} = \rho \cdot \frac{4}{3}\pi r^3
\]
kei hea \( \rho = \frac{Q}{\frac{4}{3}\pi R^3} \).
\[
q_{in} = \frac{Q}{\frac{4}{3}\pi R^3} \cdot \frac{4}{3}\pi r^3 = Q \left(\frac{r^3}{R^3}\right)
\]
– 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 kia tino whakamahia te ture a Gauss i roto i ngā tono hiko aukumetanga.
