Su'aalo iyo Doodo Tusaale ah oo ku saabsan Sharciga Gauss

Su'aalo iyo Doodo Tusaale ah oo ku saabsan Sharciga Gauss

Sharciga Gauss waa tiir muhiim ah oo ka mid ah electromagnetism-ka. Waxay bixisaa hab wax ku ool ah oo lagu xisaabiyo goobta korantada ee ka dhalata qaybinta lacagta korontada. Maqaalkan, waxaan ka hadli doonnaa dhowr dhibaato oo tusaale ah waxaanan ka wada hadli doonnaa adeegsiga Sharciga Gauss xaalado kala duwan.

Fikradda Aasaasiga ah ee Sharciga Gauss

Kahor inta aynaan bilaabin dhibaatooyinka tusaalaha ah, aan dib u eegno fikradda aasaasiga ah ee Sharciga Gauss. Sharciga Gauss wuxuu sheegayaa in wadarta qulqulka korantada \( \Phi_E \) ee ka soo baxa dusha xiran ay u dhigantaa wadarta guud ee kharashka \( q_{in} \ ) ee ku xiran dusha sare. Xisaab ahaan, Sharciga Gauss waxaa lagu qeexay sidan:

\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \]

Halkee:

– \( \Phi_E \) waa qulqulka korantada.
– \( \mathbf{E} \) waa goobta korontada.
– \( \mathbf{A} \) waa vector-ka aagga dusha sare.
– \( q_{in} \) waa dallacaadda gudaha dusha xiran.
– \( \epsilon_0 \) waa ogolaanshaha faakuumka (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \)).

Su'aal Tusaale ah 1: Goob Koronto oo ku taal Goob Kontorool oo Godan

Su'aal:
Waxaad haysataa goob godan oo gudbisa hirarka oo leh radius dibadda ah oo ah (R \) iyo wadarta guud ee dallacaadda (Q \). Go'aami goobta korantada ee ku jirta gudbiyaha godan.

Dood:
– Go'aaminta Dusha Sare ee Gaussian:
U qaado inaan dooranno dusha sare ee Gaussian wareegsan oo isku xiran oo leh gacan \( r \) gudaha godka socodsiinta (halkaas oo \( r < R \)). - Xisaabinta Dareeraha iyo Dareeraha: Maadaama gudaha godka socodsiintu uu yahay god madhan, dareeraha gudaha dusha sare ee Gaussian waa eber (\( q_{in} = 0 \)). - Adeegsiga Sharciga Gauss: Sida waafaqsan Sharciga Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \]

Maadaama \( q_{in} = 0 \), markaa qulqulka korontadu sidoo kale waa eber: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = 0 \] - Gunaanad: Maadaama qulqulka korontadu yahay eber, waxay la macno tahay in goobta korontadu \( \mathbf{E} \) meel kasta oo gudaha godka ah ay sidoo kale tahay eber. Markaa, goobta korontadu ku jirto godka socodsiinta waa \( 0 \, \text{N/C} \). Tusaale Dhibaatada 2: Goobta Korontada iyadoo la adeegsanayo Saxan aan dhammaad lahayn Dhibaatada: Xisaabi goobta korontada ee u dhow saxan bir ah oo aan dhammaad lahayn oo leh cufnaanta dallacaadda dusha sare \( \sigma \). Xalka: - Go'aaminta Dusha Sare ee Gaussian: Dooro dusha sare ee Gaussian ee qaabaysan "Gaussian pillbox" oo leh dusha sare iyo hoos saxanka, mid walbana leh bedka \( A \). - Xisaabinta Dareeraha iyo Dareeraha: Wadarta qulqulka korantada ee ka soo baxa labada daraf ee dusha sare waa: \[ \Phi_E = 2EA \] halkaasoo \( E \) uu yahay goobta korantada ee labada dhinac ee saxanka. Wadarta guud ee kharashka \( q_{in} \) oo ay ku lifaaqan tahay dusha sare ee Gaussian waa: \[ q_{in} = \sigma \cdot A \] - Adeegsiga Sharciga Gauss: Sida waafaqsan Sharciga Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \] Sidaa darteed: \[ 2EA = \frac{\sigma A}{\epsilon_0} \] Adigoo fududeynaya: \[ E = \frac{\sigma}{2\epsilon_0} \] - Gunaanad: Goobta korontada ee u dhow saxan bir ah oo aan dhammaad lahayn waa: \[ E = \frac{\sigma}{2\epsilon_0} \, \text{(N/C)} \] Tusaale Dhibaato 3: Goobta Korontada ee ku wareegsan Dhibaatada Dalacsiinta Dhibcaha: Xisaabi hadda korontada ee goobta korontada meel fog \( r \) laga bilaabo dalacsiinta dhibicda \( q \). Dood: - Go'aaminta Dusha Sare ee Gaussian: Dooro dusha sare ee Gaussian oo wareegsan oo leh radius \( r \) laga bilaabo dalacaadda barta \( q \). - Xisaabinta Daacsanaanta iyo Dalacaadda: Wadarta qulqulka korantada ee ka soo baxaya dusha sare ee Gaussian waa: \[ \Phi_E = E \cdot 4\pi r^2 \] Wadarta dallacaadda \( q_{in} \) oo ay ku lifaaqan tahay dusha sare ee Gaussian waa dalacaadda barta \( q \). - Adeegsiga Sharciga Gauss: Sida waafaqsan sharciga Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \] Markaa: \[ E \cdot 4\pi r^2 = \frac{q}{\epsilon_0} \] Adigoo fududeynaya: \[ E = \frac{q}{4\pi \epsilon_0 r^2} \] - Gunaanad: Goobta korantada ee masaafada \( r \) laga bilaabo dallacaadda dhibicda \( q \) waa: \[ E = \frac{q}{4\pi \epsilon_0 r^2} \, \text{(N/C)} \] Tusaale Su'aal 4: Goobta Korontada Gudaha iyo Dibadda Wareegga oo ka kooban dallacaadda Isku-midka ah Su'aal: Goob adag oo leh radius \( R \) waxay leedahay dallacaadda guud \( Q \) kaas oo si siman loo qaybiyo. Xisaabi goobta korantada meel ku taal gudaha kubbadda (\( r < R \)) iyo meel ka baxsan kubbadda (\( r > R \)).

Dood:

Loogu talagalay \( r < R \): - Go'aaminta Dusha Sare ee Gaussian: Dooro dusha sare ee Gaussian ee wareegsan ee radius \( r \) gudaha wareegga adag. - Xisaabinta Kharashka: Maadaama kharashku si siman u qaybsan yahay, kharashka ku jira radius \( r \) waa: \[ q_{in} = \rho \cdot \frac{4}{3}\pi r^3 \] where \( \rho = \frac{Q}{\frac{4}{3}\pi R^3} \). \[ q_{in} = \frac{Q}{\frac{4}{3}\pi R^3} \cdot \frac{4}{3}\pi r^3} = Q \left(\frac{r^3}{R^3}\right) \] - Adeegsiga Sharciga Gauss: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0} \] Markaa: \[ E \cdot 4\pi r^2 = \frac{Q \left(\frac{r^3}{R^3}\right)}{\epsilon_0} \] Adigoo fududeynaya: \[ E = \frac{Q r}{4\pi \epsilon_0 R^3} \] Markaa, goobta korantada ee ku jirta wareegga (\( r < R \)) waa: \[ E = \frac{Q r}{4\pi \epsilon_0 R^3} \] Loogu talagalay \( r > R \):

– Go'aaminta Dusha Sare ee Gaussian:
Dooro dusha sare ee Gaussian oo wareegsan oo leh radius \( r \) oo ka baxsan wareegga adag.

- Xisaabinta Culayska:
Wadarta guud ee dusha sare ee Gaussian waa wadarta guud ee xawaaraha \( Q \).

– Adeegsiga Sharciga Gauss:

\[
\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{q_{in}}{\epsilon_0}
\]

Sidaas darteed:

\[
E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0}
\]

Adigoo fududeynaya:

\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]

Markaa, garoon koronto oo ka baxsan kubbadda (\( r > R \)) waa:

\[
E = \frac{Q}{4\pi \epsilon_0 r^2}
\]

Gabagabo

Sharciga Gauss wuxuu bixiyaa qalab awood badan oo lagu falanqeeyo goobaha korontada ee xaalado kala duwan. Marka la doorto dusha sare ee Gaussian ee ku habboon oo la adeegsado mabaadi'diisa aasaasiga ah, waxaan si hufan u xisaabin karnaa qaybinta goobta korontada. Tusaalooyinka kor ku xusan, waxaan ku aragnay codsiyada sharciga Gauss xaaladaha sida goobta korontada ee wareegga hagaya, saxan bir ah oo aan dhammaad lahayn, dallac dhibic ah, iyo kubbadda oo ka kooban dallac siman. Faham iyo ku-dhaqan joogto ah ayaa hubin doona in si adag loo adeegsado sharciga Gauss codsiyada kala duwan ee electromagnetism.