Tambayoyi Misali Game da Filayen Magnetic A Kewaye da Waya Madaidaiciya
Pengantar
Filayen maganadisu koyaushe batu ne mai ban sha'awa a fannin kimiyyar lissafi, musamman a tattaunawar yadda ake samar da su da kuma yadda suke mu'amala da wutar lantarki. Wani muhimmin bangare na filayen maganadisu shine filin maganadisu da ke kewaye da waya mai ɗauke da wutar lantarki madaidaiciya. A cikin wannan labarin, za mu tattauna ainihin manufar filin maganadisu a kusa da waya madaidaiciya kuma mu samar da misalai da dama na matsaloli da mafita don zurfafa fahimtarmu.
Ka'idar Asali ta Filayen Magnetic A Kewaye da Wayoyi Madaidaiciya
Kafin mu zurfafa cikin matsalar misali, yana da mahimmanci mu fara fahimtar ka'idar filayen maganadisu a kusa da waya madaidaiciya. Dangane da dokar Biot-Savart da dokar Ampère, filin maganadisu da ke kewaye da waya madaidaiciya mai ɗauke da wutar lantarki za a iya bayyana shi ta hanyar lissafi:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
Ina,
– \( B \) shine filin maganadisu (Tesla),
– \( \mu_0 \) shine ikon shiga cikin injin \( (4\pi \sau 10^{-7} T \cdot m/A) \),
– \( I \) shine wutar lantarki da ke gudana a cikin waya (Ampere), kuma
– \( r \) shine nisan da ke tsakanin waya zuwa wurin da ake auna filin maganadisu (mita).
Wannan filin maganadisu yana samar da da'irori masu ma'ana a kewaye da wayar bisa ga ƙa'idar hannun dama. Idan babban yatsan hannu yana nuna alkiblar wutar lantarki, to yatsun da ke riƙe da wayar suna nuna alkiblar filin maganadisu.
Tambayoyi da Tattaunawa Samfura
Tambaya ta 1:
Doguwar waya madaidaiciya tana ɗauke da wutar lantarki ta 10 A. Lissafa girman filin maganadisu a nisan mita 0,2 daga wayar.
Tattaunawa:
Muna amfani da lissafin filin maganadisu da ke kewaye da waya madaidaiciya:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
An sani:
\[ I = 10 \, A \]
\[ r = 0,2 \, m \]
\[ \mu_0 = 4 \pi \sau 10^{-7} \, T \cdot m/A \]
Sauya waɗannan dabi'u cikin lissafin:
\[ B = \frac{4 \pi \sau 10^{-7} \sau 10}{2 \pi \sau 0,2} \]
\[ B = \frac{4 \pi \sau 10^{-6}}{2 \pi \sau 0,2} \]
\[ B = \frac{4 \sau 10^{-6}}{0,2} \]
\[ B = 20 \sau 10^{-6} \]
\[ B = sau 2 10^{-5} \, T \]
\[ B = 20 \, \mu T \]
Don haka, girman filin maganadisu a nisan mita 0,2 daga wayar shine 20 μT (microTesla).
Tambaya ta 2:
Wayoyi biyu masu tsayi madaidaiciya suna ɗauke da irin wannan wutar lantarki ta 5 A amma a alkibla daban-daban, kuma nisan da ke tsakaninsu mita 0,1 ne. Lissafa girman filin maganadisu a tsakiyar waya biyu.
Tattaunawa:
A tsakiyar waya biyu, nisan shine \( r = 0,05 \, m \) daga kowace waya. Muna ƙididdige filin maganadisu da ya samo asali daga waya ɗaya da farko.
Ga kowace waya:
\[ B_1 = \frac{\mu_0 I}{2 \pi r} \]
An sani:
\[ I = 5 \, A \]
\[ r = 0,05 \, m \]
\[ \mu_0 = 4 \pi \sau 10^{-7} \, T \cdot m/A \]
Sauya waɗannan dabi'u cikin lissafin:
\[ B_1 = \frac{4 \pi \sau 10^{-7} \sau 5}{2 \pi \sau 0,05} \]
\[ B_1 = \frac{4 \pi \sau 10^{-7} \sau 5}{\pi \sau 0,1} \]
\[ B_1 = \frac{20 \pi \sau 10^{-7}}{\pi \sau 0,1} \]
\[ B_1 = \frac{20 \sau 10^{-7}}{0,1} \]
\[ B_1 = 200 \sau 10^{-7} \]
\[ B_1 = sau 2 10^{-5} \, T \]
Tunda wayoyi biyu suna ɗaukar kwararar ruwa a alkibla daban-daban, filayen maganadisu suna soke juna a wannan lokacin. Jimillar filin maganadisu a wannan lokacin sifili ne.
Tambaya ta 3:
Doguwar waya madaidaiciya A tana ɗauke da wutar lantarki ta 12 A kuma tana nan a layi ɗaya da dogon waya madaidaiciya B tana ɗauke da wutar lantarki ta 8 A a daidai wannan alkibla. Kiyasta jimlar filin maganadisu a wuri mai nisan mita 0,15 daga waya A da mita 0,1 daga waya B.
Tattaunawa:
Lissafa filin maganadisu na kowace waya a wannan lokacin.
Don waya A:
\[ B_A = \frac{\mu_0 I_A}{2 \pi r_A} \]
An sani:
\[I_A = 12 \, A \]
\[ r_A = 0,15 \, m \]
Sauya darajar:
\[ B_A = \frac{4 \pi \sau 10^{-7} \sau 12}{2 \pi \sau 0,15} \]
\[ B_A = \frac{48 \pi \sau 10^{-7}}{\pi \sau 0,3} \]
\[ B_A = \frac{48 \sau 10^{-7}}{0,3} \]
\[ B_A = 160 \sau 10^{-7} \]
\[ B_A = 1,6 \sau 10^{-5} \, T \]
Don waya B:
\[ B_B = \frac{\mu_0 I_B}{2 \pi r_B} \]
An sani:
\[I_B = 8 \, A \]
\[ r_B = 0,1 \, m \]
Sauya darajar:
\[ B_B = \frac{4 \pi \sau 10^{-7} \sau 8}{2 \pi \sau 0,1} \]
\[ B_B = \frac{32 \pi \sau 10^{-7}}{\pi \sau 0,2} \]
\[ B_B = \frac{32 \sau 10^{-7}}{0,2} \]
\[ B_B = 160 \sau 10^{-7} \]
\[ B_B = 1,6 \sau 10^{-5} \, T \]
Tunda wutar lantarki a cikin wayoyi biyu tana gudana a hanya ɗaya, kuma maki suna da nisa daban-daban daga kowace waya, filin maganadisu da ke fitowa zai kasance a hanya ɗaya. Saboda haka, jimlar filin maganadisu shine jimlar waɗannan filayen maganadisu guda biyu.
\[ B_{jimla} = B_A + B_B \]
\[ B_{jimla} = 1,6 \sau 10^{-5} + 1,6 \sau 10^{-5} \]
\[ B_{jimla} = 3,2 \sau 10^{-5} \, T \]
Don haka, jimlar filin maganadisu a wannan lokacin shine 32 μT (microTesla).
Kammalawa
Fahimtar manufar filin maganadisu a kusa da waya madaidaiciya yana da matuƙar muhimmanci ga kimiyyar lissafi domin yana da aikace-aikace da yawa masu amfani. Ta hanyar amfani da misalai da tattaunawa kamar wanda ke sama, za mu iya ƙarfafa manufar asali da kuma zurfafa fahimtarmu game da yadda filayen maganadisu ke aiki a kusa da wayar da ke ɗauke da wutar lantarki. Kullum ku tuna, daidaiton nazari da fahimtar dokoki na asali suna da mahimmanci don magance matsalolin kimiyyar lissafi daban-daban.
