Nā nīnau hoʻohālike e kūkākūkā ana i ka ikaika uila
ʻO ka wehewehe ʻana o ka ikaika uila ka ikaika e kū mai ana mai ka launa pū ʻana ma waena o nā uku uila. Ua aʻo ʻia kēia ikaika ma ka physics, ʻoi aku hoʻi ma ka lālā o ka electrodynamics. Hiki ke ʻike ʻia nā hopena o ka ikaika uila i nā hanana o kēlā me kēia lā, mai ka hamo ʻana i kekahi mau mea a hiki i ka hana ʻana o nā mea uila. E kūkākūkā kēia ʻatikala i kekahi mau laʻana o nā pilikia ikaika uila a me kā lākou mau hoʻonā, a me nā manaʻo nui e pili ana i ka ikaika uila.
Hoʻolauna i ka Mana Uila
Ua hāʻawi ʻia ka ikaika uila ma waena o nā uku uila ʻelua e ke Kānāwai o Coulomb, i hōʻike ʻia e ka hoohalike:
$$ F = k_e \frac{|q_1 q_2|}{r^2} $$
Ma hea:
– ʻO \( F \) ka ikaika uila ma waena o nā uku ʻelua.
– ʻO \( q_1 \) a me \( q_2 \) nā nui o nā uku.
– ʻO \( r \) ka mamao ma waena o nā uku ʻelua.
– ʻO \( k_e \) ke kūpaʻa Coulomb (ma kahi o \(8.99 \times 10^9 \, N \cdot m^2 / C^2\)).
Ua like ke kānāwai o Coulomb me ke kānāwai o ka umekaumaha o Newton, akā he mea hoihoi mau ka umekaumaha ʻoiai hiki i ka ikaika uila ke hoihoi a hoʻowahāwahā paha ma muli o ka hōʻailona o ka hoʻopiʻi.
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: ʻElua mau uku like
Aia nā uku kiko ʻelua \( q_1 = 3 \, \mu C \) a me \( q_2 = -4 \, \mu C \) ma kahi mamao o 0.5 mika mai kekahi i kekahi. E helu i ka ikaika uila e hana ana ma waena o nā uku ʻelua.
Kūkākūkā:
1. Ka ʻike ʻana i ka ukana a me ka mamao:
– \( q_1 = 3 \, \mu C = 3 \times 10^{-6} \, C \)
– \( q_2 = -4 \, \mu C = -4 \times 10^{-6} \, C \)
– \( r = 0.5 \, m \)
2. Hoʻololi i ke kānāwai o Coulomb:
\[
F = k_e \frac{|q_1 q_2|}{r^2}
\]
\[
F = 8.99 \times 10^9 \, \frac{(3 \times 10^{-6})(4 \times 10^{-6})}{(0.5)^2}
\]
3. E helu i nā hopena:
\[
F = 8.99 \times 10^9 \, \frac{12 \times 10^{-12}}{0.25}
\]
\[
F = 8.99 \times 10^9 \times 48 \times 10^{-12}
\]
\[
F = 431.52 \times 10^{-3} \, N
\]
\[
F \approx 0.432 \, N
\]
No laila, ʻo ka ikaika uila e hana ana ma waena o nā uku ʻelua ma kahi o 0.432 Newtons.
Nīnau 2: Hoʻopiʻi ma nā uku he nui
Ua kau ʻia nā hoʻouku ʻekolu kiko, \( q_1 = 2 \, \mu C \), \( q_2 = 3 \, \mu C \), a me \( q_3 = -1 \, \mu C \) ma kahi laina pololei me ka mamao ma waena o \( q_1 \) a me \( q_2 \) he 0.6 mika a ma waena o \( q_2 \) a me \( q_3 \) he 0.4 mika. E helu i ka ikaika hopena ma \( q_2 \).
Kūkākūkā:
1. Ka ʻike ʻana i ka ukana a me ka mamao:
– \( q_1 = 2 \, \mu C = 2 \times 10^{-6} \, C \)
– \( q_2 = 3 \, \mu C = 3 \times 10^{-6} \, C \)
– \( q_3 = -1 \, \mu C = -1 \times 10^{-6} \, C \)
– ʻO ka mamao ma waena o \( q_1 \) a me \( q_2 \) he 0.6 m
– ʻO ka mamao ma waena o \( q_2 \) a me \( q_3 \) he 0.4 m
2. Hoʻoikaika iā \( q_1 \) e kūʻē iā \( q_2 \):
\[
F_{12} = k_e \frac{|q_1 q_2|}{r_{12}^2}
\]
\[
F_{12} = 8.99 \times 10^9 \, \frac{(2 \times 10^{-6})(3 \times 10^{-6})}{(0.6)^2}
\]
\[
F_{12} = 8.99 \times 10^9 \, \frac{6 \times 10^{-12}}{0.36}
\]
\[
F_{12} = 8.99 \times 10^9 \times 16.67 \times 10^{-12}
\]
\[
F_{12} \approx 0.15 \, N
\]
Aia ke kuhikuhi o ka ikaika \( F_{12} \) ma kahi mamao aku mai \( q_1 \) (ma ka ʻākau).
3. Hoʻoikaika iā \( q_3 \) e kūʻē iā \( q_2 \):
\[
F_{32} = k_e \frac{|q_3 q_2|}{r_{32}^2}
\]
\[
F_{32} = 8.99 \times 10^9 \, \frac{(1 \times 10^{-6})(3 \times 10^{-6})}{(0.4)^2}
\]
\[
F_{32} = 8.99 \times 10^9 \, \frac{3 \times 10^{-12}}{0.16}
\]
\[
F_{32} = 8.99 \times 10^9 \times 18.75 \times 10^{-12}
\]
\[
F_{32} \approx 0.17 \, N
\]
Ke huki nei ke kuhikuhi o ka ikaika \( F_{32} \) i \( q_3 \) (ma ka hema).
4. Ka Mana Hopena:
ʻOiai ʻo \( F_{12} \) ma ka ʻākau a ʻo \( F_{32} \) ma ka hema, ʻo ka hopena o ka ikaika \( F_{res} = F_{32} – F_{12} \):
\[
F_{res} = 0.17 \,N – 0.15 \,N
\]
\[
F_{res} = 0.02 \, N (ma ka hema)
\]
No laila, ʻo ka hopena o ka ikaika ma \(q_2 \) he 0.02 Newton ma ka hema.
Nīnau 3: ʻŌnaehana Hoʻouka i loko o kahi Triangle
Ua kau ʻia nā uku helu ʻekolu \( q_1 = 5 \, \mu C \), \( q_2 = -3 \, \mu C \), a me \( q_3 = 4 \, \mu C \) ma nā kihi ʻekolu o kahi huinakolu like ʻaoʻao me ka lōʻihi o ka ʻaoʻao he 0.2 mika. E helu i ka huina o ka ikaika uila e hana ana ma ka uku \( q_1 \).
Kūkākūkā:
1. Hoʻoikaika iā \( q_2 \) e kūʻē iā \( q_1 \):
\[
F_{12} = k_e \frac{|q_1 q_2|}{r^2}
\]
\[
F_{12} = 8.99 \times 10^9 \, \frac{(5 \times 10^{-6})(3 \times 10^{-6})}{(0.2)^2}
\]
\[
F_{12} = 8.99 \times 10^9 \, \frac{15 \times 10^{-12}}{0.04}
\]
\[
F_{12} = 8.99 \times 10^9 \times 375 \times 10^{-12}
\]
\[
F_{12} \approx 3.37 \, N
\]
He ʻumeʻume ke kuhikuhi o ka ikaika \( F_{12} \) ma ke kuhikuhi o \( q_2 \).
2. Hoʻoikaika iā \( q_3 \) e kūʻē iā \( q_1 \):
\[
F_{13} = k_e \frac{|q_1 q_3|}{r^2}
\]
\[
F_{13} = 8.99 \times 10^9 \, \frac{(5 \times 10^{-6})(4 \times 10^{-6})}{(0.2)^2}
\]
\[
F_{13} = 8.99 \times 10^9 \, \frac{20 \times 10^{-12}}{0.04}
\]
\[
F_{13} = 8.99 \times 10^9 \times 500 \times 10^{-12}
\]
\[
F_{13} \approx 4.50 \, N
\]
He mea hoʻopailua ke kuhikuhi o ka ikaika \( F_{13} \) mai \( q_3 \).
3. Ka Mana Hopena:
Pono mākou e hoʻohana i ke ʻano vector e ʻike ai i ka hopena o \( F_{12} \) a me \( F_{13} \), no ka mea, ʻo ke kihi ma waena o nā ikaika he 60 degere (huinakolu equilateral).
Ma ka hoʻohana ʻana i nā ʻāpana x a me y, hiki iā mākou ke hoʻoholo i ka ikaika holoʻokoʻa. Eia nō naʻe, no ka maʻalahi, e hoʻohana mākou i nā ʻano hana kiʻi a i ʻole nā helu helu e ʻike ai ʻoi aku ka nui o ka ikaika holoʻokoʻa ma mua o 4.50 N a aia ma waena o ka laina e hoʻopili ana iā \( q_2 \) a me \( q_3 \) i \( q_1 \).
Ka hopena
Ma o ka pilikia hoʻohālike a me ke kūkākūkā ʻana ma luna, hiki iā mākou ke hoʻomaopopo pehea e helu ai i ka ikaika uila me ka hoʻohana ʻana i ke Kānāwai o Coulomb. Ma ka wāwahi ʻana i ka pilikia i nā ʻanuʻu maʻalahi a me ka hoʻohana ʻana i nā noʻonoʻo vector, hiki iā mākou ke hoʻoholo i ka ikaika holoʻokoʻa ma kēlā me kēia uku i loko o ka ʻōnaehana. ʻOiai he maʻalahi ka ikaika uila i kāna ʻano makemakika, hiki ke loaʻa nā hopena koʻikoʻi a paʻakikī hoʻi i nā noi honua maoli.
