Gauss' law Definition and 341 Threads

In physics and electromagnetism , Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating to the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.
The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which forms the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa.

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  1. erick rocha

    Using Gauss's Law on two separated, and infinite plates

    Homework Statement The figure shows cross-sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with surface charge density σ = 1.06 × 10-22 C/m2. What is the y component of the electric field at points (a) above the sheets, (b) between them...
  2. KT KIM

    Derive the Analog of Gauss' law

    I am stuck with this (d) all the other (a),(b),(c) were just like (a) E= −∇φ (b) ∇ · E= −∇2φ (c) ∇ × E I know only very basic about Gauss' law, the problem is not from Electrodynamics course. I don't know what should I do to solve (d)
  3. L

    Gauss' Law & charge inside sphere

    1. Homework Statement I need to find the total charge inside the small metal sphere, inside the big metal sphere as well as outside the big metal sphere. Homework Equations What confuses me is the electric field vector. Since it's only poiting in one direction it can't originate from a...
  4. B

    Solve Gauss' Law Question: Find Max Acceleration of Two Balls

    Homework Statement Consider two balls of equal radii and masses but opposite charges, distributed uniformly over their volumes. Initially, the balls are at rest and far away from one another. Due to the Coulomb attraction, the balls start moving towards each other. The balls can be treated as...
  5. T

    Can Gauss' Law Be Applied Here?

    Homework Statement Homework Equations Gauss' Law: ∫E⋅da = qenc/ε0 E-field: E = kq/r2 The Attempt at a Solution I solved this using electric field equation, differential elements, and integration. The correct answer is, I think, E=-q/(8πε0R2) QUESTION: Can this be done via Gauss' Law? The...
  6. R

    How Does Gauss' Law Apply to an Infinite Charged Wall?

    Homework Statement [/B] Assume that the infinite wall −2 < y < 2 is filled with a uniform charge of density ρv = 3 C/m3. Obtain an expression for Ey at a general point (0,y,0) and plot Ey as a function of y over −5 < y < 5. Homework Equations E = ρv / ( 4 π ε0 ρ) ∫∫∫D⋅ds = Qenc The Attempt at...
  7. A

    Does Gauss' Law use line integrals or surface integrals?

    In my physics textbook, I see Gauss' Law as https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png\ointE dA but in other places I see it as...
  8. S

    Gauss' Law vector form problem

    Homework Statement (a) A spherical insulating shell of radius R = 3.00 m has its centre at the origin and carries a surface charge density σ = 3.00 nC/m2. Use Gauss’s law to find the electric field on the x-axis at (i) x = 2.00m and (ii) x = 4.00 m. Give you answers in the vector form. (b) A...
  9. J

    An insulating sphere inside a conducting sphere questions

    I've been given a copy of my friend's midterm exam from this same class from last term, and decided to take a crack at it to help study. One question type in particular really messes me up and it looks like the following. How would I go about solving these in the future? A solid insulating...
  10. Elvis 123456789

    Calculating Electric Field Using Gauss' Law for a Charged Insulating Slab

    Homework Statement A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x=d and x=−d. The y- and z-dimensions of the slab are very large compared to d and may be treated as essentially infinite. Let the charge...
  11. L

    Analog of Gauss' law in gravity

    Gauss law in case of sphere which has charge ##q## is ##\oint \vec{E}\cdot d\vec{S}=\frac{q}{\epsilon_0}## Is there some anologone for case of sphere with mass ##m## such that ##\oint \vec{G}\cdot d\vec{S}=4\pi \gamma m ## and what is ##\vec{G}## in that case?
  12. R

    Gauss' Law for a Nonuniform Field

    Homework Statement A closed rectangular surface with dimensions a = b and c where the faces perpendicular to the field are a*b. The left edge of the closed surface is located at position x = a, for c > a.The electric field throughout the region is nonuniform and given by E = 3*x xhat N/C...
  13. PrincessYams

    What is the electric field inside a coaxial cable with given charge densities?

    Homework Statement A coaxial cable consists of two concentric cylindrical conductors as shown here: http://puu.sh/luFZx/7f3e1ccb07.png . The inner cylinder has a linear charge density of +λ1, meaning that each meter of the cable will have λ C of charge on the inner cylinder. The outer cylinder...
  14. B

    Electric Field Inside a Dielectric-Filled Spherical Capacitor

    Homework Statement A spherical capacitor consists of two concentric spherical conductors of radii ##R_{1}## and ##R_2, (R_2 > R_1)##. The space between the two conductors is filled with a linear inhomogeneous dielectric whose relative permittivity varies with the distance ##r## from the...
  15. SnakeDoc

    Gauss' Law: Infinite plane with charge

    Homework Statement An infinite plane of charge has surface charge density 6.8 µC/m2. How far apart are the equipotential surfaces whose potentials differ by 100 V? In mm Homework Equations E=σ/(2ε0) The Attempt at a Solution So first I solved for E= 6.8e-6/(2*8.85e-12) = 384170.791 then...
  16. gracy

    Gauss' law for concentric circles

    Homework Statement A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d. The inner shell has total charge -2q and the outer shell has charge +4q. a) calculate the electric field...
  17. Q

    Gauss' law in differential form

    I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that ## \nabla\cdot E=\frac{1}{\epsilon 0}\rho##, but when I calculate it directly I get that ## \nabla\cdot E=0## (at least for ## r\neq0##). 2) Now ##...
  18. A

    Gauss' Law: Charge enclosed zero = E field zero?

    Homework Statement A cylindrical shell of radius R and length H has its charge uniformly distributed on its curved surface Find the electric field at a point P from the axis, a distance r away, measured radially outward from the midpoint of the shell such that R>r Homework Equations φ =...
  19. P

    Using Gauss' law for spherical charge distribution

    Homework Statement The Earth is constantly being bombarded by cosmic rays, which consist mostly of protons. Assume that these protons are incident on the Earth’s atmosphere from all directions at a rate of 1366. protons per square meter per second. Assuming that the depth of Earth’s atmosphere...
  20. K

    Dot product vs trigonometry in Gauss' law

    I'm currently writing my EP on various physical equations including Maxwell's equations, and I had to justify using the dot product of the normal unit vector and the electric field in the integral version. However, I can't think of a reason for not using trigonometry as opposed to the...
  21. amjad-sh

    Gauss' Law -- How did he come up with it?

    How did Gauss came out with his law εΦ=Qenc,where Φ is the flux and Qenc is the net charge inside the gaussian surface?Was it an experimental work or just a theoretical one? thanks.
  22. M

    Proof of Gauss' Law for Electrodynamics?

    I’ve been searching for a proof, using the equation for the electric field due to a moving point charge – given, for example, on page 438 of the Third Edition of David Griffith’s Introduction to Electrodynamics (equation 10.65) – that Gauss’s law holds for a moving point charge. There is no such...
  23. gimak

    Gauss' law for point charge inside sphere off center

    Homework Statement If a point charge is inside a Gaussian sphere but is off center, why is its electric field still Qenc/(e0*4*pi*r^2)?Homework Equations surface integral of E*da=Qenc/e0The Attempt at a Solution If we draw cones out from the charge. the 2 surfaces from the cones' intersection...
  24. w3dnesday

    Where is the Force on a Proton 0.79FR Relative to the Surface of a Charged Ball?

    Homework Statement Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the...
  25. B

    Exploring Gauss' Law: Understanding Spheres and Flux in Closed Surfaces

    Flux through a closed surface is zero due to external charge but why it is not so in case of sphere
  26. N

    Integral Form of Gauss' Law at Center of Finite Wire

    At the exact center of a finite wire (i.e. a distance, say $L/2$ from each end), why can I not apply Gauss's Law in integral form to find an EXACT solution for the electric field? At the center of the wire, $E$ is entirely radial, so it seems like I should be able to draw an infinitesimally...
  27. V

    Divergence Theorem Question (Gauss' Law?)

    If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
  28. Shreyansh

    Electric field inside a closed conductor.

    I recently studied Gauss' theorum according to which there is no electric field inside a closed conductor. But an isolated chage emits field lines in all possible directions. So why is it that there are no electric lines from the surface of the closed conductor inside it?
  29. H

    Gauss' Law: Calculating charge enclosed

    Homework Statement I've been doing a few Gauss' Law problems and I'm slightly confused about calculating charge enclosed by a nonconducting sphere. So I have done 2 problems that involve finding the electric field inside nonconducting spheres: 1. Charge is uniformly distributed 2. Charge per...
  30. O

    Gauss' Law & SHM Homework: Proving Simple Harmonic Motion

    Homework Statement According to an old model due to JJ Thompson, an atom consists of a cloud of positive charge within which electrons sit like plums in a pudding. THe electrons are supposed to emit light when they vibrate about their equilibrium positions in this cloud. Assume that in the case...
  31. H

    Why Doesn't Superposition Apply to Gauss's Law for Parallel Plates?

    Homework Statement Two large, flat metal plates are separated by a distance that is very small compared to their height and width. The conductors are given equal but opposite uniform surface charge densities +- \sigma. Ignore edge effects and use Gauss's law to show a) that for points far...
  32. M

    What is the Charge on a Uniformly Charged Sphere Based on Gauss' Law?

    Homework Statement Figure 23.52 gives the magnitude of the electric field inside and outside a sphere with a positive charge distributed uniformly throughout its volume. The scale of the vertical axis is set by Es = 5.0 x 10e7 N/C. What is the charge on the sphere? Homework Equations Net Flux...
  33. S

    Solve Gauss' Law Homework: Charge of Insulating Sphere & Conductor Shell

    Homework Statement A solid insulating sphere of radius 5.0 cm shown in blue has a uniform charge density throughout its volume. Concentric with this sphere is a conducting thick hollow spherical shell of inner radius 20.0 cm and outer radius 25.0 cm shown in grey. The electric field at a point...
  34. ZARATHUSTRA

    Solving Gauss' Law Confusion: q/ε & Changing Surfaces

    Ok, so a lot people said that 1. the charge outside of the enclosed surface wouldn't influence the total flux that penetrate the surface 2. q/ε wouldn't change if the charge inside the surface does not change (see my painting, so it can help you understand what i meant) i was ok with these...
  35. H

    Pointlike charge - electric potential

    This is my first time using a website like this, so I sincerely apologize if I'm posting in the wrong spot, I'm just in a bind/frustrated with this seemingly easy problem. Homework Statement What is the electric potential at a point 0.80 m away from a point charge of 3.5m C? Homework...
  36. B

    Surface Integrals and Gauss's Law

    When I learned Integrals in Calc III, the formula looked like this ∫∫ F(r(s,t))⋅(rs x rt)*dA but in physics for Gauss's law it is ∫∫E⋅nhat dA How are these the same basic formula? I know that nhat is a unit vector, so it is n/|n|, but in the actual equation, it is a dot between the cross...
  37. M

    Gauss' Law -- When can it not be applied?

    Hello everyone. I've been asked the following question by someone and I'm not quite sure which is the correct answer. I'd appreciate some help with it. The question is pretty simple: When can't be Gauss's law applied? 1. When the electric field of a point charge is dependent on the angles...
  38. G

    Finding electric field of a hemisphere (without using Gauss' Law)

    Homework Statement A charge distribution is given by ##\rho(r,\theta,phi)=\gamma r^3cos\theta,a<r,b,0\leq\theta\leq\pi/2## and is zero everywhere else. The distance from the origin ##r=\sqrt{x^2+y^2+z^2}## and ##\gamma## is a constant. Write out the electric field P along the z-axis a distance...
  39. N

    Gauss' Law - find magnitude of the electric field

    A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x=d and x=−d. The y- and z-dimensions of the slab are very large compared to d and may be treated as essentially infinite. The slab has a uniform positive charge...
  40. deep838

    Gauss' Law in a dielectric material

    This is what we have in text-books and in Wikipedia: ρ=ρb+ρf and from there we get ∇.D=ρf. But I am unable to understand why we are not considering the bound surface charge in deriving this equation. Can anyone explain this to me.
  41. davidbenari

    Gauss' Law -- is there a general proof for all geometries?

    I know how to derive Gauss's law considering only one point charge and a sphere. I've seen other derivations for other geometrical shapes and I would say this is way too tedious as a method to prove that Gauss's law always holds true. I was wondering if there is a general proof that says...
  42. R

    How Does Gauss' Law Apply to Charged Spheres?

    Hi everyone, I am struggling with gauss law. I understand the basic concept , but I can not relate it to other physics problems. I have this physics problem. I want to really understand physics, but I can't. I have watched a lo of videos but I still don't understand. I am going to give my...
  43. S

    Gauss' Law --> Electric field of an infinitely long, straight line

    Homework Statement Problem statement: An infinitely long, straight line has a uniform charge distribution of ρ C/m. Use Gauss' law to find the electric field at a point r m away from it. Solution: Consider a cylindrical volume of height ℓ with circular cross sectional area of radius r...
  44. K

    How do vectors and electric field lines differ in representing electric fields?

    I'm stuck on some high school physics, Gauss law... 1. It says this: The electric field outside the sphere is equal as if the charge of the sphere was in it's center. I don't understand this, why is this true? I get it - experiments have showed that and that's how you calculate it; but what's...
  45. F

    Q) Deriving Gauss' Law from Coulomb's Law for a Single Point Charge

    Homework Statement Q) Use Coulomb’s Law to DERIVE the Gauss Law result for the particular case of A SINGLE POINT CHARGE. That is, using Coulomb’s Law, find the ELECTRIC FLUX going through a sphere of radius r surrounding one point charge of charge-magnitude q. THANK YOU Homework...
  46. I

    How Does Gauss' Law Ensure E-field Perpendicularity on Irregular Conductors?

    For an infinite plane sheet of charge it is obvious that the E-field points directly perpendicular to the sheet. but for conductors of irregular shape. say, a wire, or even a sheet with imperfections in it, what guarantees that the E-field will point directly perpendicular from the emanating...
  47. V

    Gauss' Law and a Gaussian Sphere

    Homework Statement A Gaussian Sphere with a radius of 1m surrounds an unknown charge at the center. At this surface a uniform outward directed electric field is 1 N/C. Use Gauss' Law to calculate the amount of charge enclosed by the sphere. Homework Equations E = q/4∏εor^2 The Attempt at a...
  48. I

    How to Find Force on an Electron Using Gauss' Law?

    Homework Statement The charge density of a spherically symmetric and uniform distribution of charge is ρ = 2*10^-3 C/m^3. An electron is released one centimeter from the center. Find the magnitude and direction of the force onto the electron.Homework Equations charge density = Q/V =...
  49. B

    Is Gauss' Law Summation Necessary for Uniform Electric Fields?

    Hello, i'm doing some practice problems using Gauss' law, but I feel like my work is 'sloppy'. I'll show an example, where I think I get the right answer, but it feels like I'm neglecting to treat the summation properly, or perhaps I don;t quite understand why what I'm doing is fine...
  50. stevendaryl

    Understanding Gauss' Law for GR in Point Mass Clouds

    Suppose that you have a region of space with no fields and the only matter is in the form of a cloud of point-masses. On the one hand, within the cloud, the stress-energy tensor vanishes almost everywhere (except right at the point-mass, in which case it is infinite). So the Einstein tensor...
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