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La formula de electrostatique
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La formula de electrostatique
1.
ររៀបររៀងសាររ ើងវ ិញ
កែម ច័ន្ទរស្ម៊ី រួមគ្នា គិត រ្វើ ន្ិងទទួលខមសត្តូវ 1 រូបមន្តខ្លះៗ I. ឡាផ្លា ស 𝑂𝑝𝑒𝑟𝑎𝑡𝑒𝑢𝑟 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 a). ន្ិយមន្័យ ∇⃗⃗ = ∂ ∂x i + ∂ ∂y j + ∂ ∂z k⃗ = ( ∂ ∂x ∂ ∂y ∂ ∂z) ជាែរន្ោម ∇⃗⃗ ែាមងែូអររោរន្រេកាត ។ b). ការអន្មវតត រគអាចអន្មវតត ∇⃗⃗ បាន្ទាំងទាំហាំសាា កល ន្ិង ទាំហាំវមិចទ័រ ។ សាា កលែៈ ∇⃗⃗ . f = grad⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ f រៅថាត្កាតយង់នន្f , ជាទាំហាំវមិចទ័រ ។ វមិចទ័រែៈ ∇⃗⃗ . ∇⃗⃗ = div ∇⃗⃗ រៅថា ឌ៊ីកហវែសង់នន្ V⃗⃗ , ជាទាំហាំសាា កល។ ∇⃗⃗ ∧ V⃗⃗ = rot⃗⃗⃗⃗⃗ V⃗⃗ រៅថា រងវិល (Rotationnel )នន្ v⃗ , ជាទាំហាំសាា កល ។ សម៊ីការ Laplace ែូអររោរន្រេកាត ែូអររោរន្សម៊ីឡាាំង ែូអររោរន្កសវ៊ែ ∆𝑈 𝜕2 𝑈 𝜕𝑥 + 𝜕2 𝑈 𝜕𝑦 + 𝜕2 𝑈 𝜕𝑧 𝜕2 𝑈 𝜕𝜌2 + 1 𝜌 𝜕𝑈 𝜕𝜌 + 1 𝜌2 𝜕2 𝑈 𝜕𝜃2 + 𝜕2 𝑈 𝜕𝑧2 𝜕2 𝑈 𝜕𝑟2 + 2 𝑟 𝜕𝑈 𝜕𝑟 + 1 𝑟2 sin2 𝜑 𝜕2 𝑈 𝜕𝜃2 + 1 𝑟2 sin2 𝜑 𝜕 𝜕𝜑 ( 𝑠𝑖𝑛𝜑𝜕𝑈 𝜕𝜑 )
2.
ររៀបររៀងសាររ ើងវ ិញ
កែម ច័ន្ទរស្ម៊ី រួមគ្នា គិត រ្វើ ន្ិងទទួលខមសត្តូវ 2 II. ត្កាតយង់ (Gradient) a). ន្ិយមន្័យ ត្កាតយង់ជាអន្មគមន្៍មួយនន្ទាំហាំសាា កល។ របើ f ជាវមិចទ័រកេលភ្ជា ប់ព៊ីបកត្មបត្មួល df រៅធាតមនន្បាំលាស់ទ៊ី dl⃗⃗⃗ ។ df⃗⃗⃗ = grad⃗⃗⃗⃗⃗⃗⃗⃗⃗ f. dl⃗⃗⃗ b). សញ្ញា ណរូបវិទា បង្ហា ញព៊ីទិសរៅណាមួយនន្កេន្អគគិសន្៊ី f រែើតរ ើងរលឿន្បាំផមត។ c). ការបង្ហា ញ ែូអររោរន្រេកាត ែូអររោរន្សម៊ីឡាាំង ែូអររោរន្កសវ៊ែ 𝐸⃗ = −𝑔𝑟𝑎𝑑⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈 𝐸 𝑋 = 𝜕𝑈 𝜕𝑥 𝐸 𝑦 = 𝜕𝑈 𝜕𝑦 𝐸𝑧 = 𝜕𝑈 𝜕𝑧 𝐸𝜌 = 𝜕𝑈 𝜕𝜌 , 𝐸 𝜃 = 1 𝜌 𝜕𝑈 𝜕𝜃 𝐸 𝑍 = 𝜕𝑈 𝜕𝑧 𝐸𝑟 = 𝜕𝑈 𝜕𝑟 , 𝐸 𝜃 = 1 𝑟𝑠𝑖𝑛𝜑 𝜕𝑈 𝜕𝜃 , 𝐸 𝜑 = 1 𝑟 𝜕𝑈 𝜕𝜑 III. ឌ៊ីរវែសង់ (Divergence) a). ផាមចែាមងកេន្វមិចទ័រ ផាមចែាមងកេន្វមិចទ័រ 𝑣 ឆ្ាងកាត់នផទកែង 𝑆 គឺែាំណត់រោយ 𝜙𝑠(𝑣) = ∬ 𝑣𝑠 . 𝑑𝑠⃗⃗⃗⃗ 𝑑𝑠⃗⃗⃗⃗ ជាវមិចទ័រណ័ រម៉ា ល់រលើនផទ 𝑆 ។
3.
ររៀបររៀងសាររ ើងវ ិញ
កែម ច័ន្ទរស្ម៊ី រួមគ្នា គិត រ្វើ ន្ិងទទួលខមសត្តូវ 3 b). ន្ិយ័មន្័យ មន្ 𝑆 ជានផទបិទជមាំវិញចាំន្មច 𝑀មួយ ។ 𝑆 ែាំណត់បាន្មឌ𝑣។ 𝑑𝑖𝑣 𝑉⃗ ត្តង់ចាំន្មច 𝑀គឺ ែាំណត់រោយ៖ 𝑑𝑖𝑣 𝑉⃗ = lim 𝑣→0 ∬ 𝑣𝑠 . 𝑑𝑠⃗⃗⃗⃗ 𝑣 = lim 𝑣→0 𝜙𝑠(𝑣) 𝑣 c). សញ្ញា ណរូបវិទា អន្មគមន្៍ឌ៊ីរវែសង់ (Divergence)បង្ហា ញព៊ីវតតមន្ត្បភពកេលបាន្បរងាើតកេន្វមិចទ័រ ។ ឧទហរណ៍ ែៈ បន្ទមែរអ ិចត្ឌិចគឺជាត្បភពនន្កេន្អគគិសន្៊ី 𝐸⃗ ។ d). ការបង្ហា ញ ែូអររោរន្រេកាត ែូអររោរន្សម៊ីឡាាំង ែូអររោរន្កសវ៊ែ 𝑑𝑖𝑣⃗⃗⃗⃗⃗⃗ 𝐴 𝜕𝐴 𝑥 𝜕𝑥 + 𝜕𝐴 𝑦 𝜕𝑥 + 𝜕𝐴 𝑧 𝜕𝑥 1 𝜌 𝜕𝑈 𝜕𝜌 (𝜌𝐴 𝜌) + 1 𝜌 𝜕𝐴 𝜃 𝜕𝜃 + 𝜕𝐴 𝑧 𝜕𝑧 1 𝑟2 𝜕 𝜕𝑟 (𝑟2 𝐴 𝑟) + 1 𝑟𝑠𝑖𝑛𝜑 𝜕𝐴 𝜃 𝜕𝜃 + 1 𝑟𝑠𝑖𝑛𝜑 𝜕 𝜕𝜑 (𝑠𝑖𝑛𝜑. 𝐴 𝑧) IV. រ ៉ាូសមិន្រណៀល (Rotationnel) a). បាំលាស់ទ៊ីកេន្វមិចទ័រ មន្ 𝑣 កេន្វមិចទ័រ ន្ិង (𝑐) កខែរកាងណាមួយ ។ បាំលាស់ទ៊ីនន្ 𝑣 តាម(𝑐)គឺ 𝒞𝑐 (𝑉⃗ ) = ∫ 𝑣. 𝑑𝑙⃗⃗⃗ 𝑑𝑙⃗⃗⃗ ជាធាតមបាំលាស់ទ៊ីរលើ 𝐶
4.
ររៀបររៀងសាររ ើងវ ិញ
កែម ច័ន្ទរស្ម៊ី រួមគ្នា គិត រ្វើ ន្ិងទទួលខមសត្តូវ 4 b). ន្ិយមន្័យ មន្កខែរកាង (𝑐)បិទជមាំវិញចាំន្មច 𝑀។ (𝐶) ែាំណត់នផទ 𝑆 ។ ែាំប៉ាូសង់ណ័ រម៉ា ល់នន្ វមិចទ័រ 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ . 𝑉⃗ រលើនផទ 𝑆 ត្តង់ចាំន្មច 𝑀 គឺ ៖ (𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ . 𝑉⃗ ) 𝑛 = lim 𝑆→0 ∫ 𝑣. 𝑑𝑙⃗⃗⃗ 𝑆 = lim 𝑆→0 𝒞𝑐 (𝑉⃗ ) 𝑆 c). សញ្ញា ណរូបវិទា អន្មគមន្៍ Rotationnel បង្ហា ញព៊ីវតតមន្ចលនារាងរងវង់(ខយល់ ទឹែ វតថមរផែងៗ)កេល បរងាើតបាន្កេន្វមិចទ័រ។ ឧទហរណ៍ ែៈ ចរន្តរអ ិចត្ឌិចមន្ចលនារាងរងវង់ (Tourbillons) នន្កេន្ម៉ា រញ៉ាទិច 𝐵⃗ ។ d). ការបង្ហា ញ ែូអររោរន្រេកាត ែូអររោរន្សម៊ីឡាាំង ែូអររោរន្កសវ៊ែ 𝐸⃗ = 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ . 𝐴 𝐸 𝑋 = 𝜕𝐴 𝑧 𝜕𝑦 − 𝜕𝐴 𝑦 𝜕𝑧 𝐸 𝑦 = 𝜕𝐴 𝑥 𝜕𝑧 − 𝜕𝐴 𝑧 𝜕𝑥 𝐸𝑧 = 𝜕𝐴 𝑦 𝜕𝑥 − 𝜕𝐴 𝑥 𝜕𝑦 𝐸𝜌 = 1 𝜌 𝜕𝐴 𝑧 𝜕𝜃 − 𝜕𝐴 𝜃 𝜕𝑧 , 𝐸 𝜃 = 𝜕𝐴 𝜌 𝜕𝑧 − 𝜕𝐴 𝑧 𝜕𝜌 𝐸 𝑍 = 1 𝜌 [ 𝜕(𝜌𝐴 𝜃) 𝜕𝜌 − 𝜕𝐴 𝜌 𝜕𝜃 ] 𝐸𝑟 = 1 𝑟𝑠𝑖𝑛𝜑 [ 𝜕 𝜕𝜑 (𝐴 𝜃 𝑠𝑖𝑛𝜑) − 𝜕𝐴 𝜌 𝜕𝜃 ] 𝐸 𝜃 = 1 𝑟 [ 𝜕 𝜕𝑟 (𝑟𝐴 𝜑) − 𝜕𝐴 𝑟 𝜕𝜑 ] 𝐸 𝜑 = 1 𝑟 [ 1 𝑟𝑠𝑖𝑛𝜑 𝜕𝐴 𝑟 𝜕𝜃 − 𝜕 𝜕𝑟 (𝑟𝐴 𝜃)]
5.
ររៀបររៀងសាររ ើងវ ិញ
កែម ច័ន្ទរស្ម៊ី រួមគ្នា គិត រ្វើ ន្ិងទទួលខមសត្តូវ 5 សម្គា ល់ ៖ 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ (grad⃗⃗⃗⃗⃗⃗⃗⃗⃗ U) = 0 div(𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ A) = 0 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ 𝑈 𝐴 = 𝑔𝑟𝑎𝑑⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈 ∧ 𝐴 + 𝑈 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ 𝐴 𝑑𝑖𝑣⃗⃗⃗⃗⃗⃗ 𝑈 𝐴 = 𝐴 𝑔𝑟𝑎𝑑⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈 + 𝑈 div 𝐴 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ (𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ 𝐴) = 𝑔𝑟𝑎𝑑⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑑𝑖𝑣𝐴 − ∆𝐴 𝑑𝑖𝑣(𝐴 ∧ 𝐵⃗ ) = 𝐵⃗ 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ 𝐴 − 𝐴 𝑟𝑜𝑡⃗⃗⃗⃗⃗⃗ 𝐵⃗ សូមរែោសិទធកែកត្ប
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