Mibvunzo yemuenzaniso uye Kukurukurirana kweMutemo waGauss
Mutemo waGauss ndiwo musimboti mukuru wesimba remagetsi. Unopa nzira inoshanda yekuverenga simba remagetsi rinogadzirwa nekugoverwa kwechaji yemagetsi. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko uye tichakurukura mashandisirwo eMutemo waGauss mumamiriro akasiyana-siyana.
Pfungwa huru yeMutemo waGauss
Tisati tatanga nematambudziko emuenzaniso, ngationgororei pfungwa huru yeMutemo waGauss. Mutemo waGauss unoti kuyerera kwemagetsi kose \( \Phi_E \) kunobva panzvimbo yakavharwa kwakaenzana nechaji yose \( q_{in} \) yakavharirwa pamusoro. Pamasvomhu, Mutemo waGauss unoratidzwa se:
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \]
di mana:
– \( \Phi_E \) ndiyo nzira yemagetsi inoyerera.
– \( \mathbf{E} \) ndiyo nzvimbo yemagetsi.
– \( \mathbf{A} \) ndiyo vhekita yenzvimbo yepamusoro.
– \( q_{in} \) ndiyo chaji iri mukati menzvimbo yakavharwa.
– \( \epsilon_0 \) ndiyo mvumo yekubvisa vacuum (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \)).
Muenzaniso Mubvunzo 1: Munda weMagetsi muDunhu reConductor rine Hollow
Mubvunzo:
Une bhora rekufambisa risina chinhu rine radius yekunze \( R \) uye huwandu hwechaji \( Q \). Sarudza nzvimbo yemagetsi iri mukati me conductor isina chinhu.
Kukurukurirana:
- Kuziva kweGaussian Surface:
Ngatitii tinosarudza nzvimbo yeGaussian yakatenderera ine denderedzwa remukati memugero wemhepo (apo \(r <R \)).
- Kuverenga Flux uye Charge:
Sezvo mukati me conducting sphere iri mugomba risina chinhu, chaji iri mukati meGaussian surface iri zero (\( q_{in} = 0 \)).
- Kushandiswa kweMutemo waGauss:
Zvinoenderana neMutemo waGauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Sezvo \( q_{in} = 0 \), saka flux yemagetsi hainawo zero:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = 0
\]
- Mhedziso:
Sezvo kuyerera kwemagetsi kuri zero, zvinoreva kuti nzvimbo yemagetsi \( \mathbf{E} \) panzvimbo yega yega iri mukati megomba iri zerowo.
Saka, munda wemagetsi uri mukati megomba rekufambisa magetsi ndi \( 0 \, \text{N/C} \).
Muenzaniso 2: Munda weMagetsi neNdege Isingaperi
Mubvunzo:
Verenga nzvimbo yemagetsi iri pedyo neplate yesimbi isingaperi ine huwandu hwechaji yepamusoro \( \sigma \).
Kukurukurirana:
- Kuziva kweGaussian Surface:
Sarudza nzvimbo yeGaussian yakaita se "Gaussian pillbox" ine nzvimbo dziri pamusoro nepasi peplate, imwe neimwe ine nzvimbo \( A \).
- Kuverenga Flux uye Charge:
Kuyerera kwemagetsi kubva kumativi ese ari maviri epamusoro ndekwekuti:
\[
\Phi_E = 2EA
\]
apo \( E \) ndiyo nzvimbo yemagetsi kumativi ese eplate. Mutengo wose \( q_{in} \) wakakomberedzwa nepamusoro peGaussian ndewekuti:
\[
q_{in} = \sigma \cdot A
\]
- Kushandiswa kweMutemo waGauss:
Zvinoenderana neMutemo waGauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Kuti:
\[
2EA = \frac{\sigma A}{\epsilon_0}
\]
Nekuita kuti zvinhu zvive nyore:
\[
E = \frac{\sigma}{2\epsilon_0}
\]
- Mhedziso:
Munda wemagetsi uri pedyo neplate yesimbi isingaperi ndewekuti:
\[
E = \frac{\sigma}{2\epsilon_0} \, \text{(N/C)}
\]
Muenzaniso 3: Munda weMagetsi Wakakomberedza Poindi Imwe Cheji
Mubvunzo:
Verenga munda wemagetsi uri kure \( r \) kubva papoindi yekuchaja \( q \).
Kukurukurirana:
- Kuziva kweGaussian Surface:
Sarudza nzvimbo yeGaussian yakatenderera ine radius \(r \) yepoindi yekuchaja \(q \).
- Kuverenga Flux uye Charge:
Kuyerera kwemagetsi kunobva pamusoro peGaussian ndekwekuti:
\[
\Phi_E = E \cdot 4\pi r^2
\]
Mutengo wose \( q_{in} \) wakakomberedzwa neGaussian surface ndiyo point charge \( q \).
- Kushandiswa kweMutemo waGauss:
Zvinoenderana nemutemo waGauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Kuti:
\[
E \cdot 4\pi r^2 = \frac{q}{\epsilon_0}
\]
Nekuita kuti zvinhu zvive nyore:
\[
E = \frac{q}{4\pi \epsilon_0 r^2}
\]
- Mhedziso:
Munda wemagetsi uri kure \(r \) kubva pakuchaja kwepoindi \(q \) ndewekuti:
\[
E = \frac{q}{4\pi \epsilon_0 r^2} \, \text{(N/C)}
\]
Muenzaniso 4: Munda weMagetsi Mukati neKunze kweSphere Ine Chaji Yakafanana
Mubvunzo:
Dunhu rakasimba reradius \( R \) rine huwandu hwechaji \( Q \) hwakagoverwa zvakaenzana. Verenga nzvimbo yemagetsi iri mukati medunhu (\( r < R \)) uye kunze kwedunhu (\( r > R \)).
Kukurukurirana:
Kune \( r < R \):
- Kuziva kweGaussian Surface:
Sarudza nzvimbo yeGaussian yakatenderera ine radius \( r \) mukati me solid sphere.
- Kuverenga Mutoro:
Sezvo chaji ichigoverwa zvakaenzana, chaji iri mukati me radius \( r \) ndeiyi:
\[
q_{in} = \rho \cdot \frac{4}{3}\pi r^3
\]
apo \( \rho = \frac{Q}{\frac{4}{3}\pi R^3} \).
\[
q_{in} = \frac{Q}{\frac{4}{3}\pi R^3} \cdot \frac{4}{3}\pi r^3 = Q \left(\frac{r^3}{R^3}\right)
\]
- Kushandiswa kweMutemo waGauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Kuti:
\[
E \cdot 4\pi r^2 = \frac{Q \left(\frac{r^3}{R^3}\right)}{\epsilon_0}
\]
Nekuita kuti zvinhu zvive nyore:
\[
E = \frac{Q r}{4\pi \epsilon_0 R^3}
\]
Saka, munda wemagetsi uri mukati mebhora (\( r < R \)) ndewekuti: \[ E = \frac{Q r}{4\pi \epsilon_0 R^3} \] Kune \( r > R \):
- Kuziva kweGaussian Surface:
Sarudza nzvimbo yeGaussian yakatenderera ine radius \( r \) kunze kwebhora rakasimba.
- Kuverenga Mutoro:
Mutengo wose uri pamusoro peGaussian ndiwo mutengo wose webhora \( Q \).
- Kushandiswa kweMutemo waGauss:
\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]
Kuti:
\[
E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0}
\]
Nekuita kuti zvinhu zvive nyore:
\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]
Saka, munda wemagetsi uri kunze kwebhora (\( r > R \)) ndewekuti:
\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]
Mhedziso
Mutemo waGauss unopa chishandiso chine simba chekuongorora minda yemagetsi mumamiriro akasiyana-siyana. Nekusarudza nzvimbo yakakodzera yeGaussian uye kushandisa misimboti yayo yekutanga, tinogona kuverenga kugoverwa kweminda yemagetsi zvinobudirira. Kuburikidza nemienzaniso iri pamusoro, takaona kushandiswa kwemutemo waGauss mumamiriro ezvinhu akadai semunda wemagetsi mudenderedzwa rinofambisa magetsi, ndiro yesimbi isingaperi, charge yepoindi, uye denderedzwa rine charge yakaenzana. Kunzwisisa nekuita zvinoenderana kuchaita kuti mutemo waGauss ushandiswe zvakanaka mukushandiswa kwakasiyana-siyana kwemagetsi.
