Tambayoyi Misali Game da Filayen Wutar Lantarki akan Faranti Masu Layi
Filin lantarki a cikin faranti masu layi daya muhimmin ra'ayi ne a fannin kimiyyar lantarki kuma ana gwada shi akai-akai a matakai daban-daban na ilimi, duka a makarantar sakandare da kwaleji. Fahimtar wannan ra'ayi yana da matukar muhimmanci, domin filin lantarki a cikin faranti masu layi daya yana da alaƙa da aikace-aikace daban-daban na aiki, gami da ƙirar capacitors da sauran na'urorin lantarki. Wannan labarin zai samar da misalai na matsaloli da kuma cikakken bayani game da filin lantarki a cikin faranti masu layi daya don taimakawa fahimtar wannan ra'ayi.
Ka'idar Asali ta Filayen Wutar Lantarki a Faranti Masu Layi
Faranti masu layi biyu faranti ne masu gudanarwa guda biyu waɗanda ke ɗauke da cajin lantarki a akasin haka kuma an shirya su a layi ɗaya a wani nisa. A cikin waɗannan faranti masu layi ɗaya, cajin zai kasance a rarraba daidai a saman faranti. Dangane da dokar Gauss, filin lantarki \(E \) tsakanin faranti masu layi biyu masu layi ɗaya tare da cajin akasin haka za a iya bayyana su kamar haka:
\[ E = \frac{\sigma}{\epsilon_0} \]
Ina:
– \( \sigma \) shine yawan cajin saman da ke kan farantin,
– \( \epsilon_0 \) shine izinin injin tsabtace iska \((8.85 \times 10^{-12} \, \text{F/m})\).
A ƙarƙashin yanayi mai kyau (girman farantin yana da girma idan aka kwatanta da nisan da ke tsakaninsu), ana ɗaukar filin lantarki tsakanin farantin a matsayin iri ɗaya.
Tambayoyi da Tattaunawar Samfura
Tambaya ta 1:
An shirya faranti guda biyu masu layi daya da yanki A wanda yake da girma sosai idan aka kwatanta da nisan \(d \) tsakanin su don farantin sama ya sami caji \( +Q \) kuma farantin ƙasa yana da caji \( -Q \). Idan nisan da ke tsakanin faranti biyu shine \( 2 \, \text{mm} \), yankin kowanne faranti shine \( 1 \, \text{m}^2 \), kuma cajin \( Q \) shine \( 1 \, \mu \text{C} \), ƙayyade filin lantarki tsakanin faranti masu layi daya!
Matakan Tattaunawa:
1. Yawan Cajin Sama \( \sigma \):
Yawan cajin saman shine adadin cajin da ke cikin kowane yanki na naúrar da ke kan farantin. Ana iya ƙididdige shi ta hanyar:
\[ \sigma = \frac{Q}{A} \]
An sani:
– \( Q = 1 \, \mu \text{C} = 1 \sau 10^{-6} \text{C} \)
– \( A = 1 \, \rubutu{m}^2 \)
Don haka:
\[ \sigma = \frac{1 \times 10^{-6} \text{C}}{1 \text{m}^2} = 1 \times 10^{-6} \text{C/m}^2 \]
2. Filin Wutar Lantarki \( E \):
Filin lantarki tsakanin faranti biyu masu layi daya na cajin da ke akasin haka shine \( E = \frac{\sigma}{\epsilon_0} \).
An ba da \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \), sannan:
\[ E = \frac{1 \sau 10^{-6} \text{C/m}^2}{8.85 \sau 10^{-12} \text{F/m}} \]
\[ E = 1.13 \sau 10^5 \, \text{N/C} \]
Don haka, filin lantarki tsakanin faranti biyu shine \( 1.13 \sau 10^5 \, \text{N/C} \).
Tambaya ta 2:
Faɗin faifan faifan ya ƙunshi faranti biyu masu layi ɗaya tare da yankin \( 2 \, \text{m}^2 \) kowannensu kuma ya raba ta da nisa na \( 1 \, \text{mm} \). Idan ƙarfin da ke kan farantin sama shine \( 200 \, \text{V} \) kuma a farantin ƙasa shine 0 V, ƙayyade ƙarfin filin lantarki tsakanin faranti biyu!
Matakan Tattaunawa:
1. Lissafin Bambancin Da Ke Iya Wucewa \( V \):
An san cewa farantin saman yana da yuwuwar \( V_{\text{top}} = 200 \, \text{V} \) da farantin ƙasa \( V_{\text{bottom}} = 0 \, \text{V} \). Saboda haka, bambancin yuwuwar \( V \) tsakanin faranti biyu shine:
\[ V = V_{\text{top}} – V_{\text{bottom}} = 200 \, \text{V} – 0 \, \text{V} = 200 \, \text{V} \]
2. Ƙarfin Filin Lantarki \( E \):
Ana iya ƙididdige filin lantarki a cikin capacitor na farantin layi ɗaya ta hanyar raba bambancin yuwuwar \( V \) ta hanyar nisa \( d \):
\[ E = \frac{V}{d} \]
Idan aka ba da nisa \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \), to:
\[ E = \frac{200 \, \text{V}}{1 \times 10^{-3} \, \text{m}} = 2 \times 10^5 \, \text{V/m} \]
Don haka, ƙarfin filin lantarki tsakanin faranti biyu shine \( sau 2 10^5 \, \text{V/m} \).
Tambaya ta 3:
Mai capacitor na farantin layi ɗaya yana da faranti masu faɗin \( 0.5 \, \text{m}^2 \) da kuma tazara tsakanin faranti na \( 0.5 \, \text{cm} \). Idan wannan capacitor ɗin an caje shi da cajin \( 2 \, \mu \text{C} \), a ƙididdige ƙarfinsa da filin lantarki tsakanin faranti!
Matakan Tattaunawa:
1. Ƙarfin aiki \( C \):
Ana iya ƙididdige ƙarfin capacitor na farantin layi ɗaya ta amfani da dabarar:
\[ C = \epsilon_0 \frac{A}{d} \]
An sani:
– \( \epsilon_0 = 8.85 \sau 10^{-12} \, \text{F/m} \)
– \( A = 0.5 \, \rubutu{m}^2 \)
– \( d = 0.5 \, \text{cm} = 0.5 \sau 10^{-2} \, \text{m} \)
Don haka:
\[ C = 8.85 \sau 10^{-12} \, \text{F/m} \sau \frac{0.5 \, \text{m}^2}{0.5 \sau 10^{-2} \, \text{m}} \]
\[ C = 8.85 \sau 10^{-12} \sau 10 \, \text{F} \]
\[ C = 88.5 \sau 10^{-12} \, \text{F} \]
\[ C = 88.5 \, \rubutu{pF} \]
2. Filin Wutar Lantarki \( E \):
Domin ƙididdige filin lantarki, dole ne mu fara sanin bambancin yuwuwar \( V \). Ana iya ƙididdige bambancin yuwuwar \( V \) kamar haka:
\[V = \frac{Q}{C} \]
An sani:
– \( Q = 2 \, \mu \text{C} = 2 \sau 10^{-6} \text{C} \)
– \( C = 88.5 \sau 10^{-12} \text{F} \)
Don haka:
\[ V = \frac{2 \sau 10^{-6} \, \text{C}}{88.5 \sau 10^{-12} \, \text{F}} \]
\[ V = 22.6 \sau 10^3 \, \text{V} \]
\[ V = 22.6 \, \rubutu{kV} \]
Na gaba, ta amfani da dabarar \( E = \frac{V}{d} \), filin lantarki shine:
\[ E = \frac{22.6 \, \text{kV}}{0.5 \times 10^{-2} \, \text{m}} \]
\[ E = 22.6 \sau 10^3 \, \frac{\text{V}}{0.5 \sau 10^{-2} \, \text{m}} \]
\[ E = 4.52 \sau 10^6 \, \text{V/m} \]
Don haka, ƙarfin capacitor shine ( 88.5 \, \text{pF} \) kuma filin lantarki tsakanin faranti shine ( 4.52 \times 10^6 \, \text{V/m} \).
Kammalawa
Fahimtar filin lantarki a cikin faranti masu layi ɗaya yana da matuƙar muhimmanci ba kawai ga jarrabawar ilimi ba, har ma ga aikace-aikacen aikace-aikace a fannoni daban-daban na injiniyanci da kimiyyar lissafi. Tare da misalan matsaloli da tattaunawa a sama, ana fatan masu karatu za su sami fahimtar manufar da aikace-aikacenta, wanda zai ba su damar kusantar ta da ƙarfin gwiwa a cikin yanayi da jarrabawa na gaske.
