What is Surface: Definition and 1000 Discussions

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. Crunge

    The maximal normal force (Friction with slanted surface)

    The problem that I immediately ran into was how I would calculate N without knowing Fmax. I didn't think the y-component of N would simply be the same magnitude as mg. After being stuck for a good while I even tested if it was, by dividing the magnitude of mg with cosθ, which of course ended up...
  2. josephsanders

    I Tangential electric force at a surface

    Suppose you have an infinite plane of charge. If the surface charge density is uniform, would the tangential electric force always be zero, even if it is not a conductor nor static? My thought process for this is that if you look at each point charge and draw the electric field lines, then at...
  3. mksrm

    Voltage and electrode surface area

    Hi I am looking to find the equation that determines the minimum (and if possible maximum that might damage the electrode) voltage that starts the electrolysis process for a given area of a graphite electrode in a brine solution medium (lets say 30%) at equilibrium state. Also how does the...
  4. PeterDonis

    A Can a Null Surface Satisfy the Requirements for a Cauchy Surface?

    The definition of a Cauchy surface, as given in, for example, Wald Section 8.3, is "a closed achronal set ##\Sigma## for which ##D(\Sigma) = M##", i.e., every past and future causal curve (timelike or null) through any point in the entire spacetime intersects ##\Sigma##. The definition of...
  5. tbn032

    B Rolling of non-deforming sphere on a non-deforming rough surface?

    According to my current understanding rolling friction rolling friction is the static friction (parallel to the surface on which the object is moving) applied by the frictional surface (rough surface) on the contact point or contact area of the object whose v≠Rw(v is translational velocity and...
  6. Addez123

    Find surface of maximum flux given the vector field's potential

    The vectorfield is $$A = grad \Phi$$ $$A = x^2 + y^2 + z^2 - (x^4 + y^4 + z^4 + 2x^2y^2 + 2x^2z^2 + 2y^2z^2)$$ The surface with maximum flux is the same as the volume of maximum divergence, thus: $$div A = 6 - 20(x^2 + y^2 + z^2)$$ This would suggest at the point 0,0,0 the flux is at maximum...
  7. WMDhamnekar

    Evaluate the surface integral ##\iint\limits_{\sum} f\cdot d\sigma##

    But the answer provided is ##\frac{15}{4} ## How is that? What is wrong in the above computation of answer?
  8. hjam24

    Determine vertical velocity vector on sloped surface

    A golf is launched at a speed v,f and launch angle, β,f. The slope of the green is equal to φ. At some point the ball is located on the rim of a hole. The side view (a) and overhead view (b) looks as in the attached image.According to the author of the [paper][2] "The Physics of Putting" the...
  9. WMDhamnekar

    MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$

    Evaluate the surface integral $\iint\limits_{\sum} f \cdot d\sigma $ where $ f(x,y,z) = x^2\hat{i} + xy\hat{j} + z\hat{k}$ and $\sum$ is the part of the plane 6x +3y +2z =6 with x ≥ 0, y ≥ 0, z ≥ 0 , with the outward unit normal n pointing in the positive z direction. My attempt to answer...
  10. Ryan89

    Fatal Error The Surface Appears More Than Once in A Chain

  11. jaumzaum

    B Why is sea surface water relatively constant?

    I'm studying oceanography and the author of the book that I'm currently reading stated that sea surface water is relatively constant during the day, changing very slowly during the year. He says "this is because almost all the energy received from the sun is used in the evaporation of water"...
  12. Salmone

    I Solving a Particle on the Surface of a Sphere: Obtaining Eigenvalues

    The Hamiltonian of a particle of mass ##m## on the surface of a sphere of radius ##R## is ##H=\frac{L^2}{2mR^2}## where ##L## is the angular momentum operator. I want to solve the TISE ##\hat{H}\psi=E\psi## and in order to do that I rewrite ##L^2## in Schroedinger's representation in spherical...
  13. Salmone

    I Strange Hamiltonian of two particles on the surface of a sphere

    I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...
  14. Astronuc

    CO2 allows volcanoes to form persistent lava lakes at the surface

    Antarctica's only active volcano shows how CO2 allows volcanoes to form persistent lava lakes at the surface https://phys.org/news/2022-05-antarctica-volcano-co2-volcanoes-persistent.html
  15. A

    Problem with the positive orientation of a surface (Stokes' theorem)

    Greetings the solution is the following which I understand I do understand why the current orientation of the Path is positive regarding to stocks (the surface should remain to the left) but I don´t understand why the current N vector of the surface is positive regarding stockes theorem...
  16. Trysse

    Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##

    I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem. Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...
  17. warhammer

    Seeking Guidance to Find Surface & Volume Bound Charges of a Half Cone

    This was a trivial question I had (which I posted here on the PF EM Forum: https://www.physicsforums.com/threads/bound-charges-polarisation-of-a-half-cone.1015308/). As I received no response on the above link I decided to post the same as a self formulated HW problem. Below I have attached an...
  18. M

    Charge density on the surface of a conductor

    I have tried to solve the problem by setting as a condition that the electric field inside the conductor has to be 0, but in this way I have two unknowns (σ1 and σ2):
  19. S

    Normal vector of an embedding surface

    I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...
  20. N

    I Pentaprism with magnification, which surface is curved?

    I would like to use a pentaprism with some amount of magnification. The pentaprism will be used to reflect a real image at 90 degrees angle but I also want the reflected image to appear larger. The distance between the prism and the real image is about 70cm. The pentaprism has two reflecting...
  21. Daniel Petka

    I When is a water surface not dispersive?

    For my project, I need water waves of all frequencies to move at the same speed. I read this article, but struggled to grasp some concepts. The key idea of the article is that a raindrop hitting a water surface basically creates a pulse containing all the frequencies, and since the water is very...
  22. L

    Flux of constant magnetic field through lateral surface of cylinder

    If the question had been asking about the flux through the whole surface of the cylinder I would have said that the flux is 0, but since it is asking only about the lateral surfaces I am wondering how one could calculate such a flux not knowing how the cylinder is oriented in space. One could...
  23. BnayaMeir

    Surface Current and Electric Field

    Hi everyone! I'm pretty new in this forum, I found the topics here very relevant to my physics course. And here is my question: Given the following drawing, two infinite sheets (in y and z axis) of ideal conductive material. their thickness is infinitesimal (dx->0). The electric field is...
  24. A

    I Trying to derive Gauss' law using a cylindrical surface

    When I try to derive Gauss's law with a straight line of charge with density ##\lambda## through a cylindrical surface of length L and radius R, $$\vec E = \frac{\lambda*L}{4\pi\epsilon*r^2}$$ $$A = 2\pi*r*L$$ $$\vec E*A = \frac{\lambda *L^2}{2\epsilon*r} \neq \frac{q_{enc}}{\epsilon}$$ What am...
  25. A

    I Calculating the surface area of a sphere using dA

    Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
  26. A

    I Calculation of electric flux on trapezoidal surface

    I am confused at this calculation of the electric flux through a trapezoidal surface. The flux in should equal the flux out. The flux in equals -E*A1 where A1 is the area of the bottom of the trapezoid. The flux out equals E*A2 where A2 is the area of the top of the trapezoid. But the two fluxes...
  27. A

    A Surface impedance equation for resonators

    Hi all, I am reading this thesis and have two questions to section 5.1.5: https://rsl.yale.edu/sites/default/files/files/RSL_Theses/reagor-thesis-20151202.pdf First question is: Where did equation 5.19 come from? Second question is: If you look at this paper https://arxiv.org/pdf/1308.1743.pdf...
  28. Salmone

    I Hamiltonian of a particle moving on the surface of a sphere

    In a quantum mechanical exercise, I found the following Hamiltonian: Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian ##H=\frac{\omega}{\hbar}L^2##. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was...
  29. A

    B Rolling without slipping in non-sloping surface

    Suppose we leave an object on a non-sloping surface that rolls and moves forward without slipping. Does this object continue to move indefinitely?
  30. A

    I What happens to the buoyant force at the surface of the water?

    What happens to the buoyant force at the surface of the water for an object? The buoyant force should be greater than the weight of the object if the object were to float up but once the object floats to the surface, there is no more acceleration upwards which means the buoyant force = weight of...
  31. L

    I Surface tension and water (the smallest hole water will flow through)

    I am trying to find out what the smallest hole water will flow through. not a molecule of water, just water in general. Here is an example. I have a single walled cube that i 3d printed. When i put water in it, it leaks between the layer lines. I want to find out what the spacing between the...
  32. W

    Connection between Pressure, Surface Density and Height?

    [Mentors' note: moved from technical forums so no template] Hi All, Working on a lab write-up, and I need background equations to support the reasoning for my experiment. To outline briefly, two-part experiment, first part was finding the ideal pressure for a basketball, where I inflated it...
  33. J

    Electric Field Inside a Gaussian Surface with Point Charge q

    If I have a point charge q right outside of a gaussian surface, it makes sense that the flux is zero inside the surface because the electric field going in equals the electric field going out. However, how would the electric field be zero inside? Wouldn't it just take on the electric field of...
  34. G

    Surface charge density of a plane

    $$\phi_E=\dfrac{Q_{\textrm{enclosed}}}{\varepsilon_0}\Rightarrow Q_{\textrm{enclosed}}=9,6\cdot 10^{-7}\, \textrm{C}$$ $$Q_{\textrm{enclosed}}=\sigma S=\sigma \pi R^2\Rightarrow \sigma =\dfrac{Q_{\textrm{enclosed}}}{\pi (0,1^2)}=3,04\cdot 10^{-5}\, \textrm{C}/\textrm{m}^2$$ I have a lot of...
  35. Distern

    I Collision when a falling tank impacts the surface of a lake

    I'm writing a paper on the movie realism of the A-team movie. The basic situation is that the guys are falling inside a tank with terminal velocity into a lake. I'm stumped on how to calculate the force that is created on impact, though I imagined it being done with some kind of pulse equation...
  36. TheSpectacularSpecs

    I Surface Differential Rotation by measuring the position of sunspots

    I am doing a University lab project where I measure positions of sunspots (using images from NASA's SDO) and use them to calculate the rotation of the Sun. Currently, all is going well: I have the angular velocity of several sunspots at varying heights. However, I want to be able to find the...
  37. S

    Energy required to move an object to Earth's surface

    Energy required = ##\Delta E_p## $$\Delta E_p = -\frac{GMm}{R}+\frac{GMm}{2R}$$ $$=-\frac{1}{2} \frac{GMm}{R}$$ $$=-\frac{1}{2} \frac{GMm}{R^2} R$$ $$=-\frac{1}{2} 40R$$ $$=-20R$$ But the answer key is 40R. Where is my mistake? Thanks
  38. F

    Engineering Find the flux due to a vector crossing an open surface

    Question: Equations: My attempt: Could someone confirm my answer please?
  39. samy4408

    I don't understand this problem -- Surface area of a section of a solid sphere

    Sorry i don't understand English very well , if someone want to explain to me this problem?
  40. Addez123

    What are the limits for integrating a constrained surface with two variables?

    I start by parametarize the surface with two variables: $$r(u,v) = (u, v, \frac {d -au -bv} c)$$ The I can get the normal vector by $$dr/du \times dr/dv$$ What limits should I use to integrate this only within the elipse? I could redo the whole thing and try write r(u, v) as u being the...
  41. chwala

    Find the surface area of the water in the given prism

    My query in only on the highlighted part...c.ii. Find the question below; Find the markscheme here part c(ii) does not seem correct as i have; ##A_1=0.5 ×(0.65+0.84)0.3 ×2=0.447m^2## ##A_2 = 0.65 ×1.6=1.04m^2## ##A_3 = (0.3146 × 1.6)2=1.00672m^2## Total surface area =...
  42. A

    Problem with the orientation of a surface

    Greetings All, I have confusion to establish the orientation of a surface so please bear with me σ Those two surfaces are of course intersecting in z=10 on a circle . if I take the surface σ and I want it to be oriented so that it forms an acute angle with the axe Z , the orientation of the...
  43. A

    Max Angle of Mass on Incline Surface

    Hello, I've worked through the free-body diagram to compute the answer: tan(𝜃) = 0.67 𝜃 = arctan(0.67) = 33.822... The answer is supposed to be approximately 42. Yet, tan(42) is not 0.67, is the suggested answer wrong?
  44. diogomcs

    I need an experiment about surface tension

    Summary:: Surface tension experiment Does anyone have an idea about a SURFACE TENSION experiment to present as university class work? An experiment that is not too "simple" and repetitive (like things floating under water), and that is well designed.
  45. S

    B Surface that when superimposed takes any one point to any other point

    I don't get what is meant with the last part: that takes any one given point to any other given point. Thanks in advance
  46. A

    Difference between a closed solid and a Cartesian surface

    Greetings All! I have hard time to make the difference between the equation of a closed solid and a cartesian surface. For example in the exercice n of the exam I thought that the equation was describing a closed solid " a paraboloid locked by an inclined plane (so I thought I could use...
  47. penguin46

    How to find integrals of motion for a particle on a surface?

    I have no idea where to even start with this, please help. I barely even know what integral of motion means.
  48. H

    Bound charges of a block (top and bottom surface)

    From what I think, to find the bound charges of a block on the top and bottom surface I have to find the electric field or the displacement (D). However, I'm not sure how to proceed with a cube. For example, with a sphere ##E = \frac{Q}{4\pi \epsilon_0 r^2}## since r is constant. For a cube, it...
  49. ergospherical

    I Show Maxwell's Eqns. on a Cauchy Surface (Wald Ch. 10 Pr.2)

    This problem is Wald Ch. 10 Pr. 2.; it asks us to show that ##D_a E^a = 4\pi \rho## and ##D_a B^a = 0## on a spacelike Cauchy surface ##\Sigma## (with normal vector ##n^a##) of a globally hyperbolic spacetime ##(M, g_{ab})##. Using the expression ##E_a = F_{ab} n^b## for the electric field gives...
  50. ergospherical

    I Quasi-Static Change of Event Horizon Area

    Let ##\mathscr{H}## be a constant-##v## cross-section of the event horizon (area ##A##). The expansion is the fractional rate of change of the surface element, ##\theta = \frac{1}{\delta S} \frac{d(\delta S)}{dv}##. The problem asks to prove the formula ##\frac{dA}{dv} = \frac{8\pi}{\kappa}...
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