What is Gauss' law: Definition and 338 Discussions

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. C

    Electric field external to Conducting Hollow Sphere with charge inside

    I have read Griffiths' Chapter 2 sections on Conductors. According to it, (if I understood it correctly) if the charge is put inside the cavity of a conductor, then the equal and opposite total charge will be induced surrounding the cavity. This charge and the total charge induced surrounding...
  2. L

    Gauss' law for a cavity in an insulator

    This is a problem from Yale OCW (Shankar). The solution he gives is as follows: Sure, this makes sense. However... Superimposed rho and negative rho with radius R/2 means there is no charge enclosed in the cavity... therefore no charge -> no flux -> no electric field.
  3. M

    I Gauss' law, method of images, and force induced on conductor plates

    Hi, I would very much appreciate some guidance on the below. Consider a one-dimensional world as depicted in the attached figure. We have two (lets say positively charged particles enclosed by two conductor plates. One plate is at ##x=0##, the other at ##x=L##. The particles are at ##x_1## and...
  4. A

    Importance/Use of Gauss' Law and Ampere's Circuital Law

    So the question is like I mentioned. I know how to use GL and ACL in place of CL and BSL. I also know that they make the calculations simpler. The problem is whenever GL and ACL are used in place of CL or BSL, they are used in the following way: Gauss Law:$$\int{ E...
  5. putongren

    No Electric Charges if No Electric Field in Region

    This is a conceptual question. I think we can conclude that electric charges cannot be present if there is no electric field in that region. Is this an application of Gauss' Law? A net electric flux thru a surface indicates that there is a charge within that region. An electric field must be...
  6. F

    Magnitude of the Line Charge Density of a Power Line

    Okay so I am a little confused as to where I made a mistake. I couldn't figure out how to program Latex into this website but I attached a file with the work I did and an explanation of my thought process along the way.
  7. M

    Gauss' law in line integral, Q=##ϵ_0 ∮E.n dl=-ϵ_0 ∮∂ϕ/∂n dl##

    I know the Gauss law for surface integral to calculate total charge by integrating the normal components of electric field around whole surface . but in above expression charge is calculated using line integration of normal components of electric field along line. i don't understand this...
  8. R

    Use Gauss' Law to calculate the electrostatic potential for this cylinder

    I solved laplacian equation. and got the solution of V(r, phi) = a. +b.lnr + (summation) an r^n sin(n phi +alpha n ) + (summation) bn r ^-n sin( n phi +beta n)
  9. 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...
  10. J

    Calculating the Electric field inside an infinite planar slab using Gauss' Law

    Draw a Gaussian pill box that starts from 0 (half way between the slab) and extends towards 2 cm.$$A \times \int_{0}^{0.02} \rho dz$$ I'm not sure if I should multiply the integral by A (area) or V (volume) And if area would I multiply by 0.02^2? I'm confused here. Thanks for your help.
  11. J

    Gauss' Law applied to this Charged Spherical Shell with a small hole

    First draw a gaussian shape outside of the sphere (a larger sphere) with radius R. The total charge from the (inner) sphere will be: $$Q = \sigma A$$ $$A = 4\pi r^2$$ $$Q = \sigma 4\pi r^2$$ Use Gauss's Law to derive electric field magnitude $$\oint_{}^{} E \cdot dA = \frac{q_e}{\epsilon_o}$$...
  12. MatinSAR

    Gauss' Law in half sphere

    Picture : My answer : I guess net electric flux is 0. so electric flux passing through surface 1 = -(electric flux passing through surface 2) and electric flux passing through surface 1 is EA = E(pi)(r^2) Is it correct? Thank you ...
  13. Leo Liu

    I Gauss' law and an object with nonuniform charge distribution

    Gauss' law: $$\iint_{\partial A}\vec E\cdot d\vec A=\frac{Q}{\epsilon_0}$$ Suppose we have a unevenly charged non-conducting spherical shell, in which a Gaussian surface is placed. In this case, is the electrical field on A 0, given that there is no charge inside A? I came up with this example...
  14. bluesteels

    Quick question about which radius to use on Gauss' law problem

    confused on part A/B when I look up they did E= Q/2e(0.8)^2. But why not use the 0.100mm because that is the area of the enclosed. Same with B why did they use 100m and not 0.8m because 0.8 is smaller so it enclosed the charge
  15. bluesteels

    Electric field / Gauss' Law --

    Electric Flux = E*A = 5*6(0.05)^2. when i look up at other sources they use Electric flux = q/ (8.854*10^-12 [this is e]) equation but I am confused on why the E*A equation don't work. The answer is 0.02Nm^2/C
  16. Z

    On which objects can we apply Gauss' Law to find the electric field?

    What am I missing? I also don't get the title of the section: "Charge distributions with enough symmetry for Gauss's Law". I thought Gauss's Law was valid for any closed surface enclosing a charge. I don't understand what "enough symmetry" means in the title above. I get that with symmetry...
  17. E

    Gauss' law and flux concept issues

    Hi guys i have some concept issues about flux. My book says flux is proportional to the number of lines passing through that area. so my question is: first i am thinking that a 2D circle which has a 2r diameter and it is enclosing a 2q charge so its flux should be 2q/Epsilon.then i am thinking a...
  18. rudransh verma

    I Another way of stating Gauss' law?

    Gauss law relates the net flux phi of an electric field through a closed surface to the net charge q that is enclosed by that surface. It tells us that Phi = q/permittivity Can I say it like this : The gauss law states that the net flux of the surface depends upon the net charge enclosed by that...
  19. Leo Liu

    Gauss' law in differential form

    My book claims that the diff. form of Gauss' law is $$\nabla\cdot\mathbf E=4\pi\rho$$ Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?
  20. A

    Electric Field Shielding: Does a Conductor Shield Inside?

    Hi. I was reading about conductors in electrostatic equilibrium and how it makes sense that they have zero electric field inside the material even when an external charge is brought near. The charge density of the material just rearranges itself to cancel. Then I searched for hollow conductors...
  21. S

    Gauss' Law: Understand How to Calculate Flux

    Hello everybody To calculate the flux for the electric field I need the gauss law. There are two formula one with the integration over some area and the other is Q/e0. When do I have to use which one?
  22. G

    I Problem about the usage of Gauss' law involving the curl of a B field

    I am trying to derive that $$\nabla \times B=\mu_0 J$$ First the derivation starts with the electric field $$dS=rsin\varphi d\theta r d\varphi $$ $$ \iint\limits_S E \cdot dS = \frac{q}{4 \pi \varepsilon_0} \iint\limits_S \frac{r}{|r|^3} \cdot dS $$...
  23. cwill53

    Using the solid angle to simplify an integral when deriving Gauss' Law

    I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.
  24. wcjy

    Electric field problem using Gauss' law: Point charge moving near a line charge

    F = qE ma = (2*10^-6) * (λ / (2pi*r*ε0) ) ma = (2*10^-6) * (4*10^-6 / (2pi*4*ε0) ) => I am not certain what to put for r ( But I sub in 4 because dist is 4) a = ( (2*10^-6) * (4*10^-6 / (2pi*4*ε0) ) )/ 0.1 a = 0.35950 v^2 = U^2 + 2 a s v = 0 u^2 = -2 a s => Can't sqrt negative so...
  25. R

    Using Gauss' Law to find the field at a point

    Attached is problem 23.03 from Halliday and Resnick. We have a sphere of uniform negative charge Q = -16e and radius R = 10cm. at the center of the sphere is a positively charged particle with charge q = +5e. We are supposed to use Gauss' law to find the magnitude of the electric field at...
  26. F

    I What is the orgin of Gauss' law?

    Was Gauss' law originally derived from theory or discovered by experiment?
  27. F

    I The 'outer surface' should be redefined when applying Gauss' law?

    Gauss' law dictates that charge will only appear on the outer surface of a conductor. But if there's charge in a conducting cavity, the inner surface of the conductor will accumulate induced charge. So what's outer or inner should be redefined?
  28. E

    Gauss' law for a physical capacitor with finite thickness plates

    In derivations of capacitance it is standard to consider two oppositely charged, infinitely thin sheets. If we construct a Gaussian cylinder across one sheet, we obtain ##E_{1} = \frac{\sigma}{2\epsilon_{0}}## for one sheet, and then we can superpose this field with that from the other at an...
  29. P

    Gauss' Law question about a conducting rod

    This is my attempt, i am confused at some points a. r = 0; The Electric field is 0 b. At r = a/2.00; I verified the answer and it is non zero, but my understanding is that the net charge should be on the surface of the conductor. Hence the charge q1=5*10^-15 C, should go to the surface of the...
  30. P

    Gauss' law -- Conductor with a cavity

    I really don't understand the theory of the above kind of questions. But from the little theory i understand the Electric field is 0 inside the conductor and all the charge goes to the surface and distributes equally. a. Since the E=0 inside the conductor the point charge distributes outside...
  31. snoopies622

    I Simple case of Gauss' law for gravity

    i have little experience with the differential form of Gauss's Law, and I've tried three times now to arrive at it for a point mass M (spherically symmetric classical gravitational field) but instead of getting an answer proportional to the mass density I keep getting zero. Is the divergence...
  32. torito_verdejo

    I Gauss' Law applicability on any closed surface

    I have read multiple threads on Physics Forums, Stackexchange and Quora, as well as the explanation of Gauss Law, but still don't understand the most fundamental aspect of it: its applicability for any kind of surface. More precisely, I don't get how this follows from the fact that...
  33. E

    B Determining whether the non-integral form of Gauss' law applies

    I've just been learning about Gauss' law which as far as I can tell states that the net electric flux through a surface equals the enclosed charge divided by the permittivity of free space, and is often expressed as the integral $$\int_S {\bf{E} \cdot d \bf{A}} = \frac{Q}{\epsilon_0}$$In some...
  34. S

    I Proving Gauss' Law using a Cubical Surface

    Hi, I'm trying to prove Gauss's Law by using a cubical surface with a point charge located at its center, and I'm running up against some difficult integration. I've worked through the first integral of the surface integral, but I can't seem to figure out a proper integration technique. Here is...
  35. Jelsborg

    Inconsistency in Gauss' Law

    So the first problem stated is to show that for a charge distribution between two spherical shells of radii r1<r2, the total charge inside is described by: This is rather trivial using Gauss' law in integral form, so I regard this as completed. I have used the gradient to find the electrical...
  36. Hawkingo

    Gauss' law in Dielectrics

    [PAGE 1] [PAGE 2] [PAGE 3] so in the 2nd page,when the dielectric material is introduced the gauss's law becomes $$\oint _ { S } \vec { E } \cdot \vec { d S } = \frac { ( q - q _ { i } ) } { \epsilon _ { 0 } }$$.But my question is why the ##{ \epsilon _ { 0 } }## is in the equation.Shouldn't...
  37. N

    Find the Electric Field E using Gauss' Law

    I tried to work out both a) and b), but I am not sure if I am correct. I drew a picture with a sphere around q first with radius r and then with radius 3r. For a) ##E.A=\frac {q}{ε_°}## (when using Gauss' Law) Since ##A=4πr^2##, I substituted this in the equation and solved for E giving me...
  38. adamaero

    By using Gauss' law, can the electric field be p/(pi*ε*r^2)

    Which is better to use? The equation for the area or the circumference of a circle? Schaum's Electromagnetics (4 ed) by Edminister vs http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html
  39. M

    I Determining continuity using Gauss' law

    I know how Gauss law helps us to calculate the discontinuity at a point on the surface of a surface charge. Similarly using Gauss law, is there a way to determine the continuity at other points of electric field due to a surface charge or the continuity at all points of electric field due to a...
  40. solzonmars

    Why does a Faraday Cage work?

    Why does a Faraday Cage work? (or more generally, why does the inside of a conducting shell have no electric field if there are charges placed outside it?) I understand that this is the result of polarisation, but why does the polarisation happen to exactly cancel out the field? Could there be...
  41. Silentwhale

    Electricity and magnetism (Gauss' law)

    Homework Statement Given two things spherical shells radii r1 and r2 with r2 > r1. The inner she'll is charged uniformly with a total charge Q1, while the outer shell with Q2. A) use gauss law to computer the electric field everywhere B) Use any method to calculate the potential everywhere...
  42. E

    Gauss' Law problem: determine the electric flow through a square surface due to a nearby charge

    Homework Statement determine the electric flow through a square surface of side 2l due to a load + Q located at a perpendicular distance l from the center of the plane I really don't know how to answer this question .i need help guys Thanks Homework EquationsThe Attempt at a Solution I ended...
  43. cookiemnstr510510

    Use Gauss' Law to find E-field, tricky problem

    Homework Statement A) use gauss's Law to determine the electric field at all values of radial distance (0<r<infinity) from the center of a non-uniformly charged cylinder that is very very long and lies along the x-axis. The cylinder carries excess charge per volume ρ=[a]r^2 (the [] are supposed...
  44. cookiemnstr510510

    Determining net force using Gauss' Law

    Homework Statement We have a uniformly charged, non-conducting sphere (charge per volume,ρ, and radius, R). Then a uniformly charged ruler (charge per length,λ, and length, d) is aligned radially almost touching the surface of the sphere. Determine the net force experienced by the ruler...
  45. T

    I Why do the signs alternate when using Gauss' law for gravity?

    "Gauss's theorem can be established as follows. consider an attracting particle of mass m at the point P, and let a cone of small solid angle ω be generated by radii through P. This cone cuts the surface at the points Q1, Q2, ... taken in order from P; the parts of the surface cut off by the...
  46. cookiemnstr510510

    Conducting Sphere With Cavity E-field / Gauss' law

    Homework Statement You have a conducting sphere that is in equilibrium, it has a cavity in it with positive charge +q. If you bring another charge +q2 near the outer edge of the conductor does the total surface charge on the wall of the cavity, q(int) change? There is an image attached that...
  47. cookiemnstr510510

    Electric field sphere calculations with a hollow cavity using Gauss' law

    Homework Statement A spherical cavity is hollowed out of the interior of a neutral conducting sphere. At the center of the cavity is a point charge, of positive charge q. (picture attached) a)What is the total surface charge q(int) on the interior surface of the conductor (i.e., on the wall of...
  48. cookiemnstr510510

    Electric field using Gauss' Law

    Homework Statement Two small insulating spheres with radius 9.00×10−2m are separated by a large center-to-center distance of 0.545 m . One sphere is negatively charged, with net charge -2.35 μC , and the other sphere is positively charged, with net charge 4.35 μC . The charge is uniformly...
  49. H

    I Gauss' law applied on a four-dimensional surface

    in a 4D plane or on a four-dimensional surface can gauss law be used?