Ngā Tauira Pātai e Matapaki ana i ngā Papa Hiko i runga i ngā Pereti Whakarara
He ariā taketake te papa hiko i roto i ngā pereti whakarara i roto i te ahupūngao hiko, ā, he maha ngā whakamātautau i ngā taumata mātauranga rerekē, i te kura tuarua me te whare wānanga. He mea nui te māramatanga hōhonu ki tēnei ariā, nā te mea he hononga tata te papa hiko i roto i ngā pereti whakarara ki ngā tono mahi maha, tae atu ki te hoahoa i ngā pūnga hiko me ētahi atu taputapu hiko. Ka whakaratohia e tēnei tuhinga he tauira raruraru me te matapakinga taipitopito o te papa hiko i roto i ngā pereti whakarara hei āwhina i te māramatanga ki tēnei ariā.
Te Ariā Taketake o ngā Āpure Hiko i roto i ngā Pereti Whakarara
Ko ngā pereti whakarara he pereti kawe hiko e rua e kawe ana i ngā utu hiko rerekē, ā, e whakarite ana kia whakarara i tētahi tawhiti. I roto i ēnei pereti whakarara, ka tohatohahia te utu ki runga i te mata o ngā pereti. I runga i te ture a Gauss, ko te mara hiko \( E \) i waenganui i ngā pereti kawe hiko e rua e whakarara ana me ngā utu rerekē ka taea te whakaatu penei:
\[ E = \frac{\sigma}{\epsilon_0} \]
kāore i te mana:
– Ko te \( \sigma \) te mātotoru utu mata i runga i te pereti,
– Ko te \( \epsilon_0 \) te āheinga korehau \((8.85 \times 10^{-12} \, \text{F/m})\).
I raro i ngā āhuatanga tino pai (he nui ngā āhuatanga o ngā pereti ki te whakataurite ki te tawhiti i waenganui i a rātou), ka kiia te āpure hiko i waenganui i ngā pereti he ōrite.
Ngā Tauira Pātai me ngā Kōrerorero
Pātai 1:
E rua ngā pereti whakarara, he nui rawa te horahanga A i te tawhiti \( d \) i waenganui i a rāua, kua whakaritea kia whai utu te pereti o runga \( +Q \) me te utu te pereti o raro \( -Q \). Mēnā ko te tawhiti i waenganui i ngā pereti e rua ko \( 2 \, \text{mm} \), ko te horahanga o ia pereti ko \( 1 \, \text{m}^2 \), ā, ko te utu \( Q \) ko \( 1 \, \mu \text{C} \), tātaihia te āpure hiko i waenganui i ngā pereti whakarara!
Ngā Hipanga Kōrero:
1. Te Kīanga o te Utu Mata \( \sigma \) :
Ko te mātotoru o te mata ko te nui o te utu mō ia wae horahanga i runga i te pereti. Ka taea te tatau mā te:
\[ \sigma = \frac{Q}{A} \]
E mōhiotia ana:
– \( Q = 1 \, \mu \text{C} = 1 \times 10^{-6} \text{C} \)
– \( A = 1 \, \text{m}^2 \)
Nā reira:
\[ \sigma = \frac{1 \times 10^{-6} \text{C}}{1 \text{m}^2} = 1 \times 10^{-6} \text{C/m}^2 \]
2. Papa Hiko \( E \) :
Ko te āpure hiko i waenganui i ngā pereti e rua e whakarara ana, he rerekē te utu, ko \( E = \frac{\sigma}{\epsilon_0} \).
Ka hoatu \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \), kātahi:
\[ E = \frac{1 \times 10^{-6} \text{C/m}^2}{8.85 \times 10^{-12} \text{F/m}} \]
\[ E = 1.13 \whakareatia ki te 10^5 \, \kuputuhi{N/C} \]
Nō reira, ko te āpure hiko i waenganui i ngā pereti e rua ko \( 1.13 \times 10^5 \, \text{N/C} \).
Pātai 2:
He pūnga pereti papatahi kei roto i ngā pereti whakarara e rua, ko te horahanga o ia pereti ko \( 2 \, \text{m}^2 \) ā, ko te tawhiti e wehea ana ko \( 1 \, \text{mm} \). Mēnā ko te pūmanawa kei runga i te pereti o runga ko \( 200 \, \text{V} \) ā, ko te pereti o raro ko 0 V, tātaihia te kaha o te mara hiko i waenganui i ngā pereti e rua!
Ngā Hipanga Kōrero:
1. Te Tātaitanga o te Rerekētanga Pūmanawa \( V \):
E mōhiotia ana he pūmanawa tō te pereti o runga \( V_{\text{top}} = 200 \, \text{V} \) me te pereti o raro \( V_{\text{bottom}} = 0 \, \text{V} \). Nō reira, ko te rerekētanga pūmanawa \( V \) i waenga i ngā pereti e rua ko:
\[ V = V_{\text{runga}} – V_{\text{raro}} = 200 \, \text{V} – 0 \, \text{V} = 200 \, \text{V} \]
2. Te Kaha o te Papa Hiko \( E \):
Ka taea te tatau i te āpure hiko i roto i te pūnga pereti whakarara mā te wehewehe i te rerekētanga pūmanawa \( V \) mā te tawhiti \( d \):
\[ E = \frac{V}{d} \]
I runga i te tawhiti \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \), kātahi:
\[ E = \frac{200 \, \text{V}}{1 \times 10^{-3} \, \text{m}} = 2 \times 10^5 \, \text{V/m} \]
Nō reira, ko te kaha o te papa hiko i waenganui i ngā pereti e rua ko \( 2 \times 10^5 \, \text{V/m} \).
Pātai 3:
He pereti kei tētahi pūnga pereti whakarara, ko te horahanga he \( 0.5 \, \text{m}^2 \) me te tawhiti i waenganui i ngā pereti he \( 0.5 \, \text{cm} \). Mēnā ka utaina tēnei pūnga ki tētahi utu he \( 2 \, \mu \text{C} \), tatauhia tōna kaha me te āpure hiko i waenganui i ngā pereti!
Ngā Hipanga Kōrero:
1. Te Pūngao \( C \) :
Ka taea te tatau i te kaha o te pūnga pereti whakarara mā te whakamahi i te tātai:
\[ C = \epsilon_0 \frac{A}{d} \]
E mōhiotia ana:
– \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)
– \( A = 0.5 \, \text{m}^2 \)
– \( d = 0.5 \, \kuputuhi{cm} = 0.5 \times 10^{-2} \, \kuputuhi{m} \)
Nā reira:
\[ C = 8.85 \times 10^{-12} \, \text{F/m} \times \frac{0.5 \, \text{m}^2}{0.5 \times 10^{-2} \, \text{m}} \]
\[ C = 8.85 \times 10^{-12} \times 10 \, \text{F} \]
\[ C = 88.5 \times 10^{-12} \, \text{F} \]
\[ C = 88.5 \, \kuputuhi{pF} \]
2. Papa Hiko \( E \) :
Hei tatau i te āpure hiko, me mōhio tuatahi tātou ki te rerekētanga pūmanawa \( V \). Ka taea te tatau i te rerekētanga pūmanawa \( V \) penei:
\[ V = \frac{Q}{C} \]
E mōhiotia ana:
– \( Q = 2 \, \mu \text{C} = 2 \times 10^{-6} \text{C} \)
– \( C = 88.5 \times 10^{-12} \text{F} \)
Nā reira:
\[ V = \frac{2 \times 10^{-6} \, \text{C}}{88.5 \times 10^{-12} \, \text{F}} \]
\[ V = 22.6 \times 10^3 \, \text{V} \]
\[ V = 22.6 \, \kV} \]
Muri iho, mā te whakamahi i te tātai \( E = \frac{V}{d} \), ko te mara hiko ko:
\[ E = \frac{22.6 \, \text{kV}}{0.5 \times 10^{-2} \, \text{m}} \]
\[ E = 22.6 \times 10^3 \, \frac{\text{V}}{0.5 \times 10^{-2} \, \text{m}} \]
\[ E = 4.52 \whakareatia ki te 10^6 \, \kuputuhi{V/m} \]
Nō reira, ko te kaha o te kaha hiko ko \( 88.5 \, \text{pF} \) ā, ko te mara hiko i waenganui i ngā pereti ko \( 4.52 \times 10^6 \, \text{V/m} \).
Whakamutunga
He mea nui te mārama ki te āpure hiko i roto i ngā pereti whakarara, ehara i te mea mō ngā whakamātautau mātauranga anake, engari mō ngā tono mahi hoki i roto i ngā momo mara o te hangarau me te ahupūngao. Mā ngā tauira raruraru me ngā kōrero i runga ake nei, ko te tumanako ka pai ake te mārama o ngā kaipānui ki te ariā me tōna tono, ā, ka taea ai e rātou te whakatata atu me te māia ake ki ngā āhuatanga me ngā whakamātautau o te ao tūturu.
