Te Tātai Pūmanawa Hiko mō ngā Utu Pūwāhi Whā
Pengantar
He ariā nui te pūmanawa hiko i roto i te ahupūngao hiko e āwhina ana i a tātou ki te mārama ki te taunekeneke o ngā utu hiko i te wāhi. Ina kōrero tātou mō te utu pūwāhi, e kōrero ana tātou mō te utu e kiia ana he mea kukū ki tētahi pūwāhi kotahi i te wāhi. I roto i tēnei tuhinga, ka matapakihia e tātou ngā tātai pūmanawa hiko mō ngā utu pūwāhi rerekē e whā, me pēhea te tatau i a rātou, me ngā tono mahi o tēnei ariā.
Ariā Taketake o te Pūmanawa Hiko
Ko te pūmanawa hiko i tētahi pūwāhi i te wāhi ko te pūngao pūmanawa hiko mō ia wae utu ka pā ki tētahi utu whakamātautau pai kua whakanohoia ki taua pūwāhi. Ko te pūmanawa hiko ka inehia i roto i ngā volts (V). I roto i te pāngarau, ko te pūmanawa hiko \( V \) nā tētahi utu \( q \) i te tawhiti \( r \) mai i reira ka hoatu e te tātai:
\[ V = \frac{kq}{r} \]
Kei hea:
– Ko te pūmanawa hiko (ngaohiko) te \( V \),
– Ko te pūmau Coulomb te \( k \) (\( 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \)),
– Ko te utu (coulomb) te \( q \),
– Ko te \( r \) te tawhiti mai i te utunga ki te pūwāhi e tatauhia ai te pūmanawa (mita).
Pūmanawa Hiko o ngā Utu Pūwāhi Whā
Mena e whā ā tātou utu pūwāhi \( q_1 \), \( q_2 \), \( q_3 \), me \( q_4 \) kei ngā tūranga \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \), me \( (x_4, y_4) \) i roto i ngā taunga Cartesian, ka taea e tātou te tatau i te pūmanawa hiko katoa i tētahi pūwāhi \( P(x, y) \) mā te tāpiri i ngā pūmanawa hiko e tika ana mō ia utu i taua pūwāhi.
Ko te pūmanawa hiko katoa \( V \) i te pūwāhi \( P \) e homai ana e:
\[ V = V_1 + V_2 + V_3 + V_4 \]
Kei hea:
– Ko te pūmanawa hiko o \( V_1 \) nā \( q_1 \),
– Ko te pūmanawa hiko o \( V_2 \) nā \( q_2 \),
– Ko te pūmanawa hiko o \( V_3 \) nā \( q_3 \),
– Ko te pūmanawa hiko o \( V_4 \) nā \( q_4 \).
Ka taea te tuhi i te pūmanawa hiko e pā ana ki ia utu i te pūwāhi \( P \) penei:
\[ V_1 = \frac{k q_1}{r_1}, \quad V_2 = \frac{k q_2}{r_2}, \quad V_3 = \frac{k q_3}{r_3}, \quad V_4 = \frac{k q_4}{r_4} \]
Kei hea:
– Ko te tawhiti i waenganui i te utunga (q1) me te pūwāhi (P) ko te r_1.
– Ko te tawhiti i waenganui i te utunga (q2) me te pūwāhi (P) ko te r_2.
– Ko te tawhiti i waenganui i te utunga (q3) me te pūwāhi (P) ko te r_3.
– Ko te \( r_4 \) te tawhiti i waenganui i te utu \( q_4 \) me te pūwāhi \( P \).
Ka taea te tatau i te tawhiti \( r \) i waenganui i ngā pūwāhi e rua i roto i ngā taunga Cartesian mā te whakamahi i te tātai:
\[ r = \sqrt{(x – x_i)^2 + (y – y_i)^2} \]
Kei hea:
– Ko \( (x, y) \) ngā taunga o te pūwāhi \( P \),
– Ko \( (x_i, y_i) \) ngā taunga utu \( q_i \) (i = 1, 2, 3, 4).
Nō reira, ka taea e tātou te tatau i te tawhiti \( r \) mō ia utunga, kātahi ka whakamahi i te tātai pūmanawa hiko hei kimi i te pūmanawa i te pūwāhi \( P \).
Tauira Tātaitanga
Me tango he tauira tuturu me ngā utu pūwāhi e whā e whai ake nei:
– \( q_1 = 2 \, \mu \text{C} \) kei (0, 0),
– \( q_2 = -3 \, \mu \text{C} \) kei (1, 0),
– \( q_3 = 4 \, \mu \text{C} \) kei (0, 1),
– \( q_4 = -1 \, \mu \text{C} \) kei (1, 1).
E hiahia ana mātou ki te tatau i te pūmanawa hiko i te pūwāhi \( P \) kei (2, 2).
Tuatahi, ka tatauhia e mātou te tawhiti i waenganui i te pūwāhi \( P \) me ia utunga:
\[ r_1 = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{8} = 2\sqrt{2} \]
\[ r_2 = \sqrt{(2-1)^2 + (2-0)^2} = \sqrt{5} \]
\[ r_3 = \sqrt{(2-0)^2 + (2-1)^2} = \sqrt{5} \]
\[ r_4 = \sqrt{(2-1)^2 + (2-1)^2} = \sqrt{2} \]
Kātahi ka whakamahia e mātou tēnei uara tawhiti hei tatau i te pūmanawa hiko e pā ana ki ia utu i te pūwāhi \( P \):
\[ V_1 = \frac{8.99 \times 10^9 \times 2 \times 10^{-6}}{2\sqrt{2}} \approx 3.18 \times 10^3 \, \text{V} \]
\[ V_2 = \frac{8.99 \times 10^9 \times (-3) \times 10^{-6}}{\sqrt{5}} \approx -3.81 \times 10^3 \, \text{V} \]
\[ V_3 = \frac{8.99 \times 10^9 \times 4 \times 10^{-6}}{\sqrt{5}} \approx 7.62 \times 10^3 \, \text{V} \]
\[ V_4 = \frac{8.99 \times 10^9 \times (-1) \times 10^{-6}}{\sqrt{2}} \approx -6.36 \times 10^3 \, \text{V} \]
Ko te pūmanawa hiko katoa i te pūwāhi \( P \) ko te tapeke o ēnei pūmanawa katoa:
\[ V = 3.18 \times 10^3 – 3.81 \times 10^3 + 7.62 \times 10^3 – 6.36 \times 10^3 \approx 0.63 \times 10^3 \, \text{V} \]
Ngā Whakamahinga o te Pūmanawa Hiko
He mea nui te mārama ki te pūmanawa hiko o tētahi utunga pūwāhi i roto i ngā momo tono maha, tae atu ki:
– Te hoahoa ara iahiko hiko: Me mārama ngā kaihangarau ki te tohatoha pūmanawa i roto i te ara iahiko kia tika ai te mahi a ngā wāhanga.
– Ngā āpure hiko i roto i te koiora: He mahi tā te pūmanawa hiko ki te mahi a ngā pūtau io me te tuku tohu i roto i te tinana.
– Te tukatuka rauemi: Ka whakamahia te pūmanawa hiko i roto i ngā tikanga hiko-pūmau pērā i te whakatakotoranga hiko-pūmau me te whakamahine rauemi.
Whakamutunga
Ko te tatau i te pūmanawa hiko o ētahi utu pūwāhi me mārama ki te mahi a te pūmanawa hiko me te pānga o te tawhiti i waenga i ngā utu ki a ia. Mā tēnei ariā, ka taea e tātou te whakamārama me te hoahoa i ngā pūnaha e uru ana ki ngā taunekeneke hiko. He taputapu nui te pūmanawa hiko e āwhina ana i a tātou ki te mārama ki te ao ahupūngao i ngā taumata moroiti me te makroskopi.
