Surface Definition and 1000 Threads

A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an object is more than "a mere geometric solid", but is "filled with, spread over by, or suffused with perceivable qualities such as color and warmth".The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several non equivalent such formalizations, that are all called surface, sometimes with some qualifier, such as algebraic surface, smooth surface or fractal surface.
The concept of surface and its mathematical abstraction are both widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. The concept also raises certain philosophical questions—for example, how thick is the layer of atoms or molecules that can be considered part of the surface of an object (i.e., where does the "surface" end and the "interior" begin), and do objects really have a surface at all if, at the subatomic level, they never actually come in contact with other objects.

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  1. J

    I Is the Surface of a Sphere Locally Flat?

    Given a certain manifold in ##R^3## I've been told that at every location ##p## it is possible to encounter a reference frame from which the metric is the euclidean at zero order from that point and its first correction is of second order. This, nevertheless does not match with the following...
  2. A

    How do surface piercing propellers work?

    We have a new boat, and it has surface piercing propellers. Half of the prop is above (out of) the water, and half is below. I can't quite understand how that actually works or why it would be better.
  3. P

    Do charges always lie on the inner surface of capacitors?

    Parallel plates A, B are 5mm apart, with charges +1C and -1C respectively. Parallel plates C, D are 2mm apart, with charges +1C and -1C respectively. Capacitor CD is slid between capacitor AB. Find the potential difference between AB. The key idea to solving this problem is to suppose that +1C...
  4. AN630078

    Rates of change: surface area and volume of a sphere

    The surface area of the sphere is 4πr^2. dr/dt is given as 3cm^-1. dS/dt=dS/dr*dr/dt Differentiating 4πr^2 is dS/dr= 8πr dS/dt=8πr*3 dS/dt=24πr Given that r=5 dS/dt=24π*5=120 π The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2 dV/dt=dV/dr*dr/dt dV/dt= 4πr^2*3...
  5. neilparker62

    Ripple Pattern on Martian surface

    See graphic below. Are these ripples just wind-formed sand dunes or the remnant of an ancient (possibly catastrophic proportion) flood event? The ripples appear to emanate from a structure to the left if you look earlier in the descent video.
  6. A

    Calculating the surface tension using the method of Lecomt de Nouy

    Hello! I am having unexpected difficulties with this problem.A friend showed me an article regarding the Lecomde ring method,and at the bottom was this little problem that to solve.I've done it like this. $$ y = \frac{F}{4\pi r} $$ where F is the given force and r is = 0,0325 in meters. The...
  7. A

    Why does the top surface of a Silicon Die measure close to 0 ohms?

    I have gotten access to a large bare silicon die (almost 1" across, 14nm process) that my company gets from a fab. I've been monkeying around with an ohmmeter placed at the top of the die, with the test leads placed at various points around the die, and I almost always measure the same value...
  8. M

    Build a surface normal vector (I use Mathematica)

    Not HW, but seems to fit here. I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$ I use the following in Mathematica: r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]] z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]] x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s...
  9. F

    Whether charges can stay inside wires or on the surface of a conductor

    There's an isolated capacitor initially carrying nonzero net charge. Then place the capacitor in a circuit connected with ideal wires (no resistance). Where will the excess charge go? Can they stay in wires or on any surfaces of conductors in the circuit? Electric field needs to be 0 everywhere...
  10. H

    Oscillation of a particle on a parabolic surface [equation of motion]

    Hi, I have a particle on a parabolic surface $$y = Ax^2$$ and I have to show that the frequency is $$\omega = \sqrt{2Ag}$$ I don't know how to deal with a parabola. I don't think I can use the polar coordinates like a circle. I don't see how to start this problem and in which coordinates...
  11. J

    Stainless Steel track surface grade to match a 440C roller?

    I have a machine I am designing that has a carriage supported by Track rollers on a machined flatbar track. The load that each individual roller will experience could reach 2000lbs. worst case scenario. The track surface is machined 304L - 1" wide as you can see in the green in the attached...
  12. S

    B Darkest Place on the Surface of the Earth

    Hello All On mid-winter's day, where is the darkest place in the northern hemisphere, and at what time? I'm assuming no moon, no northern lights, and no local light pollution. Would it be at the North Pole? best regards ... Stef
  13. A

    What is the relationship between surface tension and forces?

    Hello! To this I did what was recommended and this is what it looks like $$ F = mg $$ $$ F = \rho * V * g $$ $$ F = \ rho * \pi^2 * h * g $$ Than for the surface tension I did the same thing to get an expression for F. $$ y = \frac {F} {2 \pi r}$$ Than tried to get F out and than...
  14. J

    Typical surface roughness heights for aircraft surfaces

    Hello! I am investigating some preliminary aerodynamics on a regional turboprop and for the drag model the surface roughness height of the airplane is required as an input. For this i do not have any data. In which range are the typical values for modern aircraft surfaces? Thanks a lot!
  15. Martian2020

    B Why is air more dense closer to the Earth's surface? - particle's view

    Could not find reason on molecular level. E.g. here some explanation "There are two reasons: at higher altitudes, there is less air pushing down from above, and gravity is weaker farther from Earth's center." However, when I start to imagine particles, when there are only a few I get opposite...
  16. S

    Force at the end of a Rod at rest on an inclined surface

    Question diagram, attempt at solution below I need to cancel some of the terms in the moment equation but a not sure which ones to start with. I don’t know μ so can not calculate FA, so should probably substitute FA = RB2.
  17. Omega0

    B Inverse Transformation from Response Surface

    Let us say we have data which is for simplicity in N tables. All the tables have the same number of rows and columns. The columns ##A_i## have for all tables the same meaning (say measured quantaties like pressure, temperature) where the first 3 columns is the position in space. Again for...
  18. EnergyInMotion

    Reflection of Radio Waves by a Mirror-like surface

    Hey guys, I'm new to the physicsforums. I wanted to share some videos I made and see if anyone was interested or wanted to discuss what they see. In this video I show in a manner in which is VERY easy to see, that a reflective surface will reflect not just light but also radio waves (and most...
  19. S

    Can a diamond's color be simulated by only a surface coloring?

    I was reading this article about the a replica of the famous Hope Diamond and its antecedents: https://www.nytimes.com/2020/11/28/style/hope-diamond-story-smithsonian-copy.html This part seems to say that the color is to be simulated by a coating: It would seem to me that the proper attenuation...
  20. L

    Magnetic Flux through a Closed Surface and Maxwell's laws

    Hi everyone, I have a question about Maxwell's laws. According to Maxwell the magnetic flux of a magnetic field through a close surface is 0. But his third law says the circuitation of an electric field depends from magnetic flux variation. I can't understand how this can be possible since...
  21. Adams2020

    I The surface area of an oblate ellipsoid

    In "An Introduction to Nuclear Physics by W. N. Cottingham, D. A. Greenwood" for the surface area of an oblate ellipsoid, the following equation is written for small values of ε : The book has said this without proof. I found the following formula for the desired shape: No matter how hard I...
  22. Monoxdifly

    MHB What Is the Minimum Surface Area for a Cuboid Box with a Square Base?

    The volume of a cuboid box with a square base is 2 litres. The production cost per unit of its top and its bottom is twice the production cost per unit of its lateral sides. Suppose the side length of its base is x and the height of the cuboid is h. The minimum production cost is reached when...
  23. N

    What is the Correct Equation for Velocity on a Curved Surface?

    Here's my solution: I've tried to find the equation of vx. But the graph that it is generating is not right. I am not able to figure out what is wrong in the equation for vx. Please let me know where is the equation wrong and what is wrong? Here's the graph:
  24. D

    Lens selection for close proximity surface photos

    Good day all, I am searching for some advice on how I could successfully take photos of a surface that is 5mm away from a CMOS camera sensor. My ideas so far are; 1.) Find a lens with sub millimeter focal length (0.2mm by my calculations) 2.) Utilize mirrors to increase object distance 3.) Pin...
  25. TheGreatDeadOne

    Using the Divergence Theorem on the surface of a sphere

    The integral that I have to solve is as follows: \oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r' Then I apply the divergence theorem, resulting in: \iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
  26. A

    Analytical Mechanics: Regularity Conditions on Constraint Surface

    Hi, In my course in analytical mechanics, it is said that for a system of n particles subjected to r constraint equations, it is necessary to impose regularity conditions on the constraint surface defined by G = 0 where G is a function of the position of the position of the particles and time...
  27. A

    Vertical beam on a frictionless surface

    This could also be posted in the Math / differential equations sub, but it also involves the derivation which is classical physics. So I was doubting :smile:. So, I'm dusting off my dynamics a bit and found this problem of a thin beam on a frictionless surface in a different forum and decided...
  28. WhiteWolf98

    Surface Pressure Coefficient Distribution of a Doublet in a Uniform Flow

    So, it's a long way to the solution, but I'm finding it difficult to find a starting point. I'm going to say that as a first step, I should find what the value of the stream function ##\psi## is, at the surface. In order to do this, I need to use the following equation: ##F=\phi+i\psi## If I...
  29. andyonassis

    Finding the angle between a plumb line and the Earth's surface.

    From Newton's second law: $$T_{x} = F_{turn}$$ So $$T \sin \theta = ma$$ $$T_{y} = F_{y}$$ so $$T \cos \theta = mg$$ Equate the two equations to get: $$ \frac{T \sin \theta}{a} = \frac{T \cos \alpha}{g} $$ and the angle is given by: $$tan (\theta) = \frac{a}{g} $$ where ##r = \frac{v}{w}## and...
  30. Mr_Allod

    Bound Bulk Current Density and Surface Current Density

    Hi there, I've worked through most of this question but I'm stuck on the final part, showing that total bulk current ##I_B## is equal and opposite to total surface current ##I_S##. I calculated ##\vec H## the normal way I would if I was looking for ##\vec B## in an infinitely long cylindrical...
  31. F

    Water flow rate and drip surface tension

    hello i have a drip tube with water dripping and when i increase the flow rate or frequency of drips they get bigger/more massive. i see the equation mass x gravity = 3.14(tube diameter)(surface tension) my mass is changing but not gravity, 3.14 or tube diameter so surface tension must be...
  32. Hamiltonian

    B Doubt on the derivation of an equation for a surface integral

    this method of derivation is approximating the function using a polyhedron. concentrating on one of the surfaces(say the L'th surface which has an area ##\Delta S_l## and let ##(x_l,y_l,z_l)## be the coordinate of the point at which the face is tangent to the surface and let ##\hat n## be the...
  33. K

    Surface integral: Calculate the heat flow from a cylinder

    Hi, I am trying to calculate the heat flow across the boundary of a solid cylinder. The cylinder is described by x^2 + y^2 ≤ 1, 1 ≤ z ≤ 4. The temperature at point (x,y,z) in a region containing the cylinder is T(x,y,z) = (x^2 + y^2)z. The thermal conductivity of the cylinder is 55. The...
  34. rbmartel

    I Which area should I use to calculate the force on a submerged surface?

    Let's say we have a tank filled with water only half way up. I want to calculate the force being applied by the liquid on one of the walls, that's F = P.A. For the area (A), should I consider the area of the entire wall (H.L), or only the area of the wall that's in contact with the liquid...
  35. S

    Strange pattern on a concrete paved surface

    I found this funny pattern in a slightly damp concrete paved parking area. Any ideas what it is?Detail :
  36. CatalinaAlmendra

    Chemistry How Does Surface Area Affect the Free Energy of a Liquid?

    Hello, so first of all I want to clarify that english is not my first lenguage, so I'm really sorry for possible future errors. Second, this is a problem from my physical chemistry class, and I'm not sure where it fits better, if here or in the physics homework help, I'm sorry :( So, I don't...
  37. M

    Divergence Theorem Verification: Surface Integral

    Hi, I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result. Method: Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
  38. archaic

    Maximum electric field at the surface of a Van de Graaff generator

    I know that the potential of the sphere at its surface is ##V(a)=kQ/a##, and the electric field generated by it is ##E(a)=kQ/a^2##, which gives me ##V(a)=aE(a)##. When the electric field at the surface is as in the question, we have...
  39. archaic

    Finding net charge on a sphere's surface

    The electric field caused by the surface distribution on a point ##a## meters far from it is$$E(a)=\frac{kQ}{(R+a)^2}$$from which I get$$Q=\frac{(R+a)^2E(a)}{k}=\frac{(R)^2E(0)}{k}=\frac{(0.705)^2\times867}{8.99\times10^9}\approx4.79\times10^{-8}$$and I take its negative because the direction of...
  40. agnimusayoti

    Electric field a distance z from the center of a spherical surface

    Well, I really don't understand what is the use of the hint. I try to solve this problem with Coulomb's Law and try to do in spherical coordinates and got very messy infinitesimal field due to the charge of infinitesimal surface element of the sphere. Here what I got: $$\vec{r}=\vec{r_P} +...
  41. I

    I Confirming my knowledge on surface integrals

    Hi, I want to make sure my understanding of calculating surface integrals of vector fields is accurate. It was never presented this way in a textbook, but I put this together from pieces of knowledge. To my understanding, surface integrals can be calculated in four different ways (depending on...
  42. M

    How is this formula for the surface derived?

    The pages shown in the pictures are from an engineering book. I am not sure how the thickness of sphere plays a role in the inner surface of the sphere. I know that the surface area of sphere is ##4 \pi r^2 ## . Picture 2 shows how that formula plays a role in understanding the stress...
  43. A

    Problem in the parametrisation of a surface

    Good day my question is the following: How did they determine the the orientation of the first and last loop? and why in the parametrisation of the second loop we have: (3cost;-3 sint;1/2) ( the negative sign puzzles me) thanks this is the solution of the book
  44. A

    Problem figuring out the surface of integration

    Good day I have a problem figuring out the surface of integration according to the exercice, we have a paraboloid that cross a disk on the xz plane, the parabloid cross the xz plane on a smaller disk r=√3/3 so for me after going to the final step of integration and using polar coordinate i...
  45. U

    I Fixing an orientation for a connected smooth surface

    I am studying on Zorich, Mathematical Analysis II, 1st ed. pag. 174-175. After having properly explained how orientations (equivalence classes) are defined for smooth k-dimensional surfaces in ##\mathbb {R} ^ n## that can be described with a single map, move on to the more general case by...
  46. Tony Hau

    What is the meaning of electric field "immediately outside the surface"?

    Today when I am reading Griffith's electrodymamics on surface charge and force on conductors, I have come across two very ambiguous terms: electric field at the surface and immediately outside the surface. The context of these two words is as follows: The electric field immediately outside is...
  47. U

    I Calculating Surface Area of Schwarzschild Black Hole w/Weyl Coordinates

    Recently, I was tasked to find the surface area of the Schwarzschild Black Hole. I have managed to do so using spherical and prolate spheroidal coordinates. However, my lecturer insists on only using Weyl canonical coordinates to directly calculate the surface area. The apparent problem arises...
  48. brotherbobby

    Surface Tension - Using hot soapy water to wash clothes

    I mention the details in the book (verbatim) in the form of a paragraph in green below. Later I ask my questions in blue font for better reading. "Surface tension also explains why hot, soapy water is used for washing. To wash clothing thoroughly, water must be forced through the tiny spaces...
  49. S

    A How does the number of particles increase the surface adsorption?

    How the number of the particles in a solution increase the probability to get adsorption on a surface? which physical terms explain this? For example when I increase the concentration of molecules in a solution I can see that the adsorption and the aggregation on the surface happen.
  50. M

    A Clarifying question about surface states

    Recently, I was giving a presentation for my research group related to ARPES to improve my understanding. I mentioned that the probing depth of ARPES is a few angstroms, meaning we can only look at surface states, and I was sort of giving a real space picture. In other words, ARPES can only...
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