Applications of Laplace Equation in Gravitational Field.pptx

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Two students presented on the application of the Laplace equation in gravitational fields. The Laplace equation describes gravitational potential and is represented mathematically as the sum of second derivatives equaling zero. For a given gravitational potential function V, the students found the potential value at a point and showed that the function does not satisfy the Laplace equation.

Engineering

Department Of Mathematics
Mathematical Modeling

PRESENTATION
PRESENTERS
Mr. Ihsan Ullah (732)
Mr. Shahab Khan (733)
Application of Laplace Equation
In Gravitational Field
7th December, 2022

PRESENTATION
APPLICATIONS OF LAPLACE EQUATION:
It is named after the physicist Pierre-Simon Laplace.
The Laplace equation is derived to make the calculations in Physics easy
Laplace Equation is used in many branches of Physics.
such as Electrostatic,
Gravitation,
Thermal Physics
and Fluid Dynamics.
Here we only present the application of Laplace Equation in Gravitation.
Con…

APPLICATIONSOF LAPLACE EQUATION
IN GRAVITIONAL FIELD:
𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝑳. 𝑬 𝒊𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒆𝒅 𝒂𝒔
𝛁𝟐𝑽 =
𝝏𝟐𝑽
𝝏𝒙𝟐
+
𝝏𝟐𝑽
𝝏𝒚𝟐
= 𝟎 (𝒊𝒏 𝟐𝑫)
𝛁𝟐𝑽 =
𝝏𝟐𝑽
𝝏𝒙𝟐
+
𝝏𝟐𝑽
𝝏𝒚𝟐
+
𝝏𝟐𝑽
𝝏𝒛𝟐
= 𝟎 (𝒊𝒏 𝟑𝑫)
PRESENTATION
Con…

PRESENTATION
1.SOME FACTS FROM PHYSICS:
If 𝛻2𝑉 = 0, The Laplace equation in electrostatics is defined for electric
potential V.
If g = −𝛻𝑉 then 𝛻2
𝑉 = 0, the Laplace equation in gravitational field.
If 𝛻2
𝑢 = 0, u is the velocity of the steady flow.
Con…

PRESENTATION
Let 𝑔 be the gravitational field, 𝜌 be the mass density, and 𝐺 the
gravitational constant.
Then Gauss law for gravitation in differential form is 𝛻 ∙ 𝑔 = −4𝜋𝐺𝜌
The gravitational field is conservative and can therefore be expressed
as the negative gradient of gravitational potential:
𝑔 = −𝛻𝑉
𝛻 ∙ 𝑔 = 𝛻 ∙ −𝛻𝑉 = 𝛻2
𝑉
⇒ 𝛻2 𝑉 = −𝛻 ∙ 𝑔
Con…

PRESENTATION
Using differential form of Gauss Law of gravitation
⇒ ∇2
V = 4πGρ
Which is the Poisson’s Equation for gravitational fields.
In empty space ρ = 0 and we have
𝛻2
𝑉 = 0
which is Laplace equation in gravitational fields.
Con…

PRESENTATION
QUESTION:
Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational
potential at P and also verify whether the G.P satisfy the Laplace
equation?
SOLUTION:
Given V = 4𝑥2
𝑦𝑧2
at 𝑃(1,2,1), the gravitational potential will be
V = 4 1 2 2 1 2 = 8
Now Let us verify the Laplace equation
Con…

At the END
SpecialThanks
To
Dear Sir: Dr.NIGAR ALI SAHIB
AND
ALL CLASSMATES

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Applications of Laplace Equation in Gravitational Field.pptx

2. Department Of Mathematics Mathematical Modeling

3. PRESENTATION PRESENTERS Mr. Ihsan Ullah (732) Mr. Shahab Khan (733) Application of Laplace Equation In Gravitational Field 7th December, 2022

4. PRESENTATION APPLICATIONS OF LAPLACE EQUATION: It is named after the physicist Pierre-Simon Laplace. The Laplace equation is derived to make the calculations in Physics easy Laplace Equation is used in many branches of Physics. such as Electrostatic, Gravitation, Thermal Physics and Fluid Dynamics. Here we only present the application of Laplace Equation in Gravitation. Con…

5. APPLICATIONSOF LAPLACE EQUATION IN GRAVITIONAL FIELD: 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝑳. 𝑬 𝒊𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒆𝒅 𝒂𝒔 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 (𝒊𝒏 𝟐𝑫) 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 + 𝝏𝟐𝑽 𝝏𝒛𝟐 = 𝟎 (𝒊𝒏 𝟑𝑫) PRESENTATION Con…

6. PRESENTATION 1.SOME FACTS FROM PHYSICS: If 𝛻2𝑉 = 0, The Laplace equation in electrostatics is defined for electric potential V. If g = −𝛻𝑉 then 𝛻2 𝑉 = 0, the Laplace equation in gravitational field. If 𝛻2 𝑢 = 0, u is the velocity of the steady flow. Con…

7. PRESENTATION Let 𝑔 be the gravitational field, 𝜌 be the mass density, and 𝐺 the gravitational constant. Then Gauss law for gravitation in differential form is 𝛻 ∙ 𝑔 = −4𝜋𝐺𝜌 The gravitational field is conservative and can therefore be expressed as the negative gradient of gravitational potential: 𝑔 = −𝛻𝑉 𝛻 ∙ 𝑔 = 𝛻 ∙ −𝛻𝑉 = 𝛻2 𝑉 ⇒ 𝛻2 𝑉 = −𝛻 ∙ 𝑔 Con…

8. PRESENTATION Using differential form of Gauss Law of gravitation ⇒ ∇2 V = 4πGρ Which is the Poisson’s Equation for gravitational fields. In empty space ρ = 0 and we have 𝛻2 𝑉 = 0 which is Laplace equation in gravitational fields. Con…

9. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Given V = 4𝑥2 𝑦𝑧2 at 𝑃(1,2,1), the gravitational potential will be V = 4 1 2 2 1 2 = 8 Now Let us verify the Laplace equation Con…

10. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐 𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: We have 𝝏𝟐𝑽 𝝏𝒙𝟐 = 𝟖𝒚𝒛𝟐 = 𝟖 𝟐 𝟏 𝟐 = 𝟏𝟔 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 𝝏𝟐𝑽 𝝏𝒙𝒛 = 𝟖𝒙𝟐 𝒚 = 𝟖 𝟏 𝟐 𝟐 = 𝟏𝟔 Use these values, we have 𝝏𝟐 𝑽 𝝏𝒙𝟐 + 𝝏𝟐 𝑽 𝝏𝒚𝟐 + 𝝏𝟐 𝑽 𝝏𝒛𝟐 = 𝟏𝟔 + 𝟎 + 𝟏𝟔 = 𝟑𝟐 ≠ 𝟎 Con…

11. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Hence the gravitational potential at 𝑷(𝟏, 𝟐, 𝟏) is 8 and it doesn’t satisfy the Laplace Equation.

12. At the END SpecialThanks To Dear Sir: Dr.NIGAR ALI SAHIB AND ALL CLASSMATES

13.

14. PRESENTATION PRESENTERS Mr. Ihsan Ullah (732) Mr. Shahab Khan (733) Application of Laplace Equation In Gravitational Field 7th December, 2022

15. PRESENTATION APPLICATIONS OF LAPLACE EQUATION: It is named after the physicist Pierre-Simon Laplace. The Laplace equation is derived to make the calculations in Physics easy Laplace Equation is used in many branches of Physics. such as Electrostatic, Gravitation, Thermal Physics and Fluid Dynamics. Here we only present the application of Laplace Equation in Gravitation. Con…

16. APPLICATIONSOF LAPLACE EQUATION IN GRAVITIONAL FIELD: 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝑳. 𝑬 𝒊𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒆𝒅 𝒂𝒔 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 (𝒊𝒏 𝟐𝑫) 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 + 𝝏𝟐𝑽 𝝏𝒛𝟐 = 𝟎 (𝒊𝒏 𝟑𝑫) PRESENTATION Con…

17. PRESENTATION 1.SOME FACTS FROM PHYSICS: If 𝛻2𝑉 = 0, The Laplace equation in electrostatics is defined for electric potential V. If g = −𝛻𝑉 then 𝛻2 𝑉 = 0, the Laplace equation in gravitational field. If 𝛻2 𝑢 = 0, u is the velocity of the steady flow. Con…

18. PRESENTATION Let 𝑔 be the gravitational field, 𝜌 be the mass density, and 𝐺 the gravitational constant. Then Gauss law for gravitation in differential form is 𝛻 ∙ 𝑔 = −4𝜋𝐺𝜌 The gravitational field is conservative and can therefore be expressed as the negative gradient of gravitational potential: 𝑔 = −𝛻𝑉 𝛻 ∙ 𝑔 = 𝛻 ∙ −𝛻𝑉 = 𝛻2 𝑉 ⇒ 𝛻2 𝑉 = −𝛻 ∙ 𝑔 Con…

19. PRESENTATION Using differential form of Gauss Law of gravitation ⇒ ∇2 V = 4πGρ Which is the Poisson’s Equation for gravitational fields. In empty space ρ = 0 and we have 𝛻2 𝑉 = 0 which is Laplace equation in gravitational fields. Con…

20. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Given V = 4𝑥2 𝑦𝑧2 at 𝑃(1,2,1), the gravitational potential will be V = 4 1 2 2 1 2 = 8 Now Let us verify the Laplace equation Con…

21. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐 𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: We have 𝝏𝟐𝑽 𝝏𝒙𝟐 = 𝟖𝒚𝒛𝟐 = 𝟖 𝟐 𝟏 𝟐 = 𝟏𝟔 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 𝝏𝟐𝑽 𝝏𝒙𝒛 = 𝟖𝒙𝟐 𝒚 = 𝟖 𝟏 𝟐 𝟐 = 𝟏𝟔 Use these values, we have 𝝏𝟐 𝑽 𝝏𝒙𝟐 + 𝝏𝟐 𝑽 𝝏𝒚𝟐 + 𝝏𝟐 𝑽 𝝏𝒛𝟐 = 𝟏𝟔 + 𝟎 + 𝟏𝟔 = 𝟑𝟐 ≠ 𝟎 Con…

22. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Hence the gravitational potential at 𝑷(𝟏, 𝟐, 𝟏) is 8 and it doesn’t satisfy the Laplace Equation.

23. At the END SpecialThanks To Dear Sir: Dr.NIGAR ALI SAHIB AND ALL CLASSMATES

24.

25. PRESENTATION PRESENTERS Mr. Ihsan Ullah (732) Mr. Shahab Khan (733) Application of Laplace Equation In Gravitational Field 7th December, 2022

26. PRESENTATION APPLICATIONS OF LAPLACE EQUATION: It is named after the physicist Pierre-Simon Laplace. The Laplace equation is derived to make the calculations in Physics easy Laplace Equation is used in many branches of Physics. such as Electrostatic, Gravitation, Thermal Physics and Fluid Dynamics. Here we only present the application of Laplace Equation in Gravitation. Con…

27. APPLICATIONSOF LAPLACE EQUATION IN GRAVITIONAL FIELD: 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝑳. 𝑬 𝒊𝒔 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒆𝒅 𝒂𝒔 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 (𝒊𝒏 𝟐𝑫) 𝛁𝟐𝑽 = 𝝏𝟐𝑽 𝝏𝒙𝟐 + 𝝏𝟐𝑽 𝝏𝒚𝟐 + 𝝏𝟐𝑽 𝝏𝒛𝟐 = 𝟎 (𝒊𝒏 𝟑𝑫) PRESENTATION Con…

28. PRESENTATION 1.SOME FACTS FROM PHYSICS: If 𝛻2𝑉 = 0, The Laplace equation in electrostatics is defined for electric potential V. If g = −𝛻𝑉 then 𝛻2 𝑉 = 0, the Laplace equation in gravitational field. If 𝛻2 𝑢 = 0, u is the velocity of the steady flow. Con…

29. PRESENTATION Let 𝑔 be the gravitational field, 𝜌 be the mass density, and 𝐺 the gravitational constant. Then Gauss law for gravitation in differential form is 𝛻 ∙ 𝑔 = −4𝜋𝐺𝜌 The gravitational field is conservative and can therefore be expressed as the negative gradient of gravitational potential: 𝑔 = −𝛻𝑉 𝛻 ∙ 𝑔 = 𝛻 ∙ −𝛻𝑉 = 𝛻2 𝑉 ⇒ 𝛻2 𝑉 = −𝛻 ∙ 𝑔 Con…

30. PRESENTATION Using differential form of Gauss Law of gravitation ⇒ ∇2 V = 4πGρ Which is the Poisson’s Equation for gravitational fields. In empty space ρ = 0 and we have 𝛻2 𝑉 = 0 which is Laplace equation in gravitational fields. Con…

31. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Given V = 4𝑥2 𝑦𝑧2 at 𝑃(1,2,1), the gravitational potential will be V = 4 1 2 2 1 2 = 8 Now Let us verify the Laplace equation Con…

32. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐 𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: We have 𝝏𝟐𝑽 𝝏𝒙𝟐 = 𝟖𝒚𝒛𝟐 = 𝟖 𝟐 𝟏 𝟐 = 𝟏𝟔 𝝏𝟐𝑽 𝝏𝒚𝟐 = 𝟎 𝝏𝟐𝑽 𝝏𝒙𝒛 = 𝟖𝒙𝟐 𝒚 = 𝟖 𝟏 𝟐 𝟐 = 𝟏𝟔 Use these values, we have 𝝏𝟐 𝑽 𝝏𝒙𝟐 + 𝝏𝟐 𝑽 𝝏𝒚𝟐 + 𝝏𝟐 𝑽 𝝏𝒛𝟐 = 𝟏𝟔 + 𝟎 + 𝟏𝟔 = 𝟑𝟐 ≠ 𝟎 Con…

33. PRESENTATION QUESTION: Let 𝐕 = 𝟒𝒙𝟐𝒚𝒛𝟐 at a given point 𝑷(𝟏, 𝟐, 𝟏), then find the gravitational potential at P and also verify whether the G.P satisfy the Laplace equation? SOLUTION: Hence the gravitational potential at 𝑷(𝟏, 𝟐, 𝟏) is 8 and it doesn’t satisfy the Laplace Equation.

34. At the END SpecialThanks To Dear Sir: Dr.NIGAR ALI SAHIB AND ALL CLASSMATES

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