What is Gauss's law: Definition and 383 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. Heisenberg7

    B Electric Field Created by a Finite Plate

    Let's assume that we have a finite plate which is at the center of a cartesian coordinate system. Now let's define a point ##r## with coordinates ##(0, 0, z)##. My question is, can we use Gauss's law to find the electric field at this point? The direction of the electric field is going to be up...
  2. cianfa72

    I Are Maxwell's equations linearly dependent?

    HI, consider the 4 Maxwell's equations in microscopic/vacuum formulation as for example described here Maxwell's equations (in the following one assumes charge density ##\rho## and current density ##J## as assigned -- i.e. they are not unknowns but are given as functions of space and time...
  3. N

    Electric field at a point inside a non-uniformly charged sphere

    My solution is this: $$q = \varepsilon_0 \int E.dA$$ Based on gauss's law. Taking the derivative of both sides with respect to $$A$$ we get: $$\frac{dq}{dA} = \varepsilon_0 E$$ From chain rule: $$\frac{dq}{dA} = \frac{\frac{dq}{dr}}{\frac{dA}{dr}}$$ On the other hand: $$q = \int \rho dv = \int...
  4. flyusx

    I Thought I Understood Gauss' Law (Sheets)

    We know Gauss's law for an infinite sheet yields ##\textbf{E}=\frac{\sigma}{2\varepsilon_{0}}##. This is relatively elementary and I completely understand the derivation. This is also valid when looking at a parallel plate capacitor (the electric field is additive between the plates yielding...
  5. S

    Gauss' law and Faraday cage problem

    Is this a good response? The lift is a conductor, therefore electrons can move freely. The charges on a conductor reside on the outer surface as they like to be as far from each other as they possibly can be due to the repulsive coulomb force. There is no charge between the inner and the outer...
  6. MatinSAR

    Finding electric field between two conducting plates using Gauss' law

    This picture is from Sears and Zemansky's University Physics. It considers ##S_1## as a gaussian surface then it finds electric field between two plates. The only thing that I cannot understand is why it doesn't consider the electric field due to negative charges on other plate. Then electric...
  7. J

    Using Gauss's Law to Calculate Charge Density Function

    I've attached what I have so far. Used Gauss's law, everything seemed to make sense except the units don't work out in the end. The charge density function if given by: r(z)=az, where z is the perpendicular distance inside the plane.
  8. dliu1004

    Question on a Gauss's Law problem

    Summary:: I understand the basics of Gauss's Law and how to solve some of the simpler problems, but I cannot seem to solve these two questions. For question 007, one of my friends told me I had to ignore the outer shell? I did that: I integrated rho dV: (6.02*r*pi*r^2*h) from r=0 to r=.0462...
  9. mohammed

    Solving Electric Field Vector Problems Using Gauss's Law

    I'm preparing for exam but it seems I can't find problems similar to this on the internet. Here I will apply Gauss's law on the electric field vector to get the charge density. but the problem is that I can't find similar examples on the internet that uses direct vectors on Maxwell's equations...
  10. V

    Electric Flux for a cube problem

    I have tried to understand the solution given in the book which is as pasted below. The solution uses Gauss's Law but makes no mention of which Gaussian surface is used. The diagram that I have used to understand this problem is also given at the end. From the diagram, faces OADG, OABE and OEFG...
  11. V

    Gauss's Law application in Electrostatics

    (a) Due to Coulomb's law all charges whether internal or external to Gaussian surface will contribute to the electric field. This is also mentioned as it's correct answer. (b) The answer is "equal to", which makes no sense to me. It could be greater than, equal to, or less than that obtained...
  12. H

    Find ##E_0## and ##k## for ##E= E_0 \sin(k r -\omega t)## using Gauss

    ##\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}## ##\vec{E}_0 k cos(kr -\omega t) = \frac{\rho}{\epsilon_0}## ##E_0 = \frac{\rho}{\epsilon_0} / k cos(kr -\omega t)## and ##k^2 = (\arccos{\frac{\rho}{E_0 \epsilon_0}} + \omega t)/r## I don't think it makes sense since I found ##k = \pm...
  13. preachingpirate24

    Electric Field inside the material of a hollow conducting sphere

    Let's say I place a positive point charge inside a hollow conducting sphere. If we take a Gaussian surface through the material of the conductor, we know the field inside the material of the conductor is 0, which implies that there is a -ve charge on the inner wall to make the net enclosed...
  14. K

    What Causes Bubble Oscillation According to Gauss's Law?

    From Gauss's Law give ##E=\dfrac{\sigma}{2\epsilon_0}## ##\therefore P_e=\dfrac{\sigma^2}{2\epsilon_0}## Consider at equilibrium (before bubble being charged) ##P_i=P_0+\dfrac{4S}{R}## Using Newton's 2nd Law ##\Sigma F=m\ddot{R}## Let ##R+\delta R## be the new radius Give (after binomial...
  15. P

    Understanding Gauss's Law in Cylindrical Shells of Non-Infinite Length

    My question is going to be rather specific. I am trying to understand how Gauss's law applies to this scenario. I know if a cylindrical shell is infinitely long, and there is an external electric field, the inside of the shell will have an electric field of zero everywhere. I am wondering...
  16. K

    Electric field using Gauss's Law, but in open cylinder given only r?

    I have no idea how to approach the problem using Gauss's Law. I found the electric field using superposition, and it was incorrect. I am assuming you treat the wire as a continuous electric field, and then also treat the pipe as a continuous electric field. I solved for this using...
  17. Hawkingo

    What is the effect of the dielectric material on Gauss' law?

    [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...
  18. J

    Gauss's law -- Integral form problem

    Problem Statement: The effective charge density of the electron cloud in a hydrogen atom in its quantum mechanical ground state turns out to be given by pnot(e^-(r/rnot)), where pnot is a negative constant (the clouds charge density at r=0) and rnot is a constant (rnot=0.025nm). Use gauss's law...
  19. L

    Flux Through a Cube's Face with our Point Charge at a Corner

    Homework Statement A charge q is placed at one corner of a cube. What is the value of the flux of the charge's electric field through one of its faces? Homework Equations The flux surface integral of an electric field is equal to the value of the charge enclosed divided by the epsilon_naught...
  20. T

    How Do You Calculate Charge Density in a Uniform Sphere?

    Homework Statement A charge Q is uniformly distributed throughout a nonconducting sphere of radius R. Write the expression of the charge density in the sphere? Homework Equations Charge density ρ=dQ/dV Gauss's Law ∫EdA = E(4ϖr^2) The Attempt at a Solution If Q is uniform then ρ=Q/dV and the...
  21. E

    I Time harmonic case of Gauss's Law

    In a chapter building up to the theory of plane waves my book starts by introducing time harmonic electric fields and defines a special case of Gauss's law.curl(H) = J + dD/dt curl(H) = sigma * E + epsilon * dE/dt if E is time harmonic and spacially dependent... E(x,y,z,t) let E' represent the...
  22. Avinash Verma

    Solving Gauss's Law Problem: Two +q Charges Separated by d

    Consider two point charges +q,+q. Separated by distance d. now there exist a point P on the line joining these two charges where electric field cancels out at distance d/2 from the charge. If we make a Gaussian surface at this point and work out the surface integral it won't be zero. since two...
  23. nivamani Rajbongshi

    I Gauss's Law Application: Electric Field Intensity on Non-Conducting Sheets

    why electric field intensity at a point of a non conducting sheet is independent of r?
  24. r0ss

    I Why can't we apply Gauss's law to a circular disk?

    We apply Gauss's law to find electric field at a point due to chaged plane or plate. But what's wrong when applying to circular disk which can also be considered as a plane?
  25. R

    I Understanding Gauss's Law and Special Cases in Maxwell's Equations

    I have an issue with Gauss's Law when it is considered one equation of Maxwell's complete system of equations. I don't have an issue with it when it is a standalone equation, but when it is one of several equations put together to form a complete system of equations, there is an issue. How is...
  26. A

    I Solving the Basel problem using Gauss's law

    Hi, so I came across this video: which shows an interesting way to solve the Basel problem using lighthouses. Imagine a lighthouse that has absolute brightness 1. The apparent brightness then follows an inverse-square law. Now imagine an infinite number line with positive integers only (and...
  27. M

    Charge distributions of two infinite parallel plates

    Homework Statement Two infinitely large conducting plates with excess charge 2Q and 3Q are placed parallel to one another, and at a small distance from one another. How are the charges 2Q and 3Q distributed? You may assume that infinitely large sheets of charge produce electric fields that are...
  28. G

    Using Gauss's Law to Calculate Electric Field of a Charged Thread

    Hello, Can not Gauss's Law be used to calculate the electric field generated by a uniformly charged finite thread? I suppose it is because I can not consider the electric field constant (always going to the same direction), and for this I would have to do it by parts (the lateral flow, and the...
  29. T

    Rectifying my logic of Gauss's Law

    Homework Statement Homework Equations The Attempt at a Solution E4Πr2 = Q/∈0 49000⋅4Π4.12 =Q/∈0 Q = 91.6 μC Qshell = Q = 91.6 μC Qshell = Qinner + Qouter 91.6 = -3.3 + Qouter Qouter = 94.9 μC Can someone point out the error? I have skipped too many lectures and I am catching up right...
  30. S

    Apparent disagreement between Coulomb's Law and Gauss' Law

    This is probably my misunderstanding, so please clarify. In a region of empty space, there are two point charges with the charges+Q and -Q. Exactly in the middle of the two charges (distance r from both charges) is point P, colinear with the centers of both charges. A Gaussian surface that...
  31. G

    Gauss's Law and Parallel Line Charges

    Homework Statement Find an equation for the net electric field at a point, above and between, two infinite line charges, one with line charge density λ and the second with line charge density -λ. The point is a distance R from both line charges, a distance y above the midpoint between charges...
  32. Matt Chu

    Potential Energies of Two Charged Cylinders

    Homework Statement Problem 1.24 (this is unimportant; it's just a different way of calculating the potential energy of a solid cylinder) gives one way of calculating the energy per unit length stored in a solid cylinder with radius a and uniform volume charge density ##\rho##. Calculate the...
  33. Matt Chu

    Angle of escaping electric field lines

    Homework Statement Two charges 2q and -q are located at x = 0 and x = a respectively. There are field lines extending from the positive charge and lines going inwards to the negative charge. Some of these lines go from the positive charge to the negative, but some go off to infinity from the...
  34. Tom MS

    Why Does Diatomic Hydrogen Form Despite Zero Net Electric Field?

    Say you had two isolated hydrogen atoms. Because of the spherical distribution of electronic charge on each hydrogen and the net charge of 0 outside each atom, wouldn't Gauss's law dictate a 0 net electric field outside each atom? If this is the case, why does diatomic hydrogen so readily form...
  35. J

    Surface bound charge on the outer surface of a dielectric

    Homework Statement With regards to a one dimensional conducting wire with a homogeneous charge density λ surrounded by a cylindrical glass dielectric of radius R, find: (a). The displacement vector inside the dielectric (b). The surface bound charges on the surface of the dielectric Sorry...
  36. NFuller

    I Understanding Gauss's Law: Where Does the Argument Break Down?

    So, this has been bothering me for a few days and I'm having trouble understanding where the fault is. If we consider a uniform charge density ##\rho## extending through all space, then by symmetry, I would argue that ##\mathbf{E}=0## in all space. However, this does not agree with what a naive...
  37. P

    I Gauss's law of sphere using integral

    Hey I was just practicing Gauss's law outside a sphere of radius R with total charge q enclosed. So I know they easiest way to do this is: ∫E⋅da=Q/ε E*4π*r^2=q/ε E=q/(4*πε) in the r-hat direction But I am confusing about setting up the integral to get the same result I tried ∫ 0 to pi ∫0 to...
  38. C

    Calculating Electric Field Using Gauss's Law for a Spherical Charge Distribution

    Homework Statement A charge of -30 μC is distributed uniformly throughout a spherical volume of radius 10.0 cm. Determine the electric field due to this charge at a distance of (a) 2.0 cm, (b) 5.0 cm, and (c) 20.0 cm from the center of the sphere. Homework Equations Eq. (1): E⋅A=qenc/ε Eq...
  39. H

    Gauss's Law to find E with non-uniform charge distribution

    Homework Statement Hello, this is more of a conceptual question than a concrete homework assignment question. I'm learning about Gauss's law and the Prof did an exercise on a sphere with uniform charge distribution, where he found E(r). The trick was, that E(r) was constant over the Gaussian...
  40. E

    Finding electric potential between two concentric spheres.

    Homework Statement A spherical shell with inner radius A and outer radius 3A which has a uniform charge density, i.e charge per unit volume, p0. Find difference in electric potential between the center of the shell and a point a distance 2A from the center. Homework Equations The answer given...
  41. G Cooke

    Potential on the inner surface of a spherical shell

    Is there a potential on the inner surface of a charged spherical shell? I know that there is no electric field on the inner surface, as shown by Gauss's Law, but that isn't enough information to say that the potential (V) there is zero since E = dV/dr, so V could be a nonzero constant. If...
  42. F

    Gauss's law and symmetric charge distributions

    Having read several introductory notes on Gauss's law, I have found it very frustrating that when the author comes to discussing the standard examples, in which one considers symmetric charge distributions, they do not explicitly discuss the symmetries of the situation, simply stating that, "by...
  43. Swapnil Das

    Evaluation of Surface Integral in Gauss's Law

    I am a tenth grader, and a newbie to Advanced Calculus. While working out problems sets for Gauss's Law, I encountered the following Surface Integral: I couldn't attempt anything, having no knowledge over surface integration. So please help.
  44. I

    Electric Flux of Two Hemispheres with different radii

    Homework Statement Two hemispherical surfaces, 1 and 2, of respective radii r1 and r2, are centered at a point charge and are facing each other so that their edges define an annular ring (surface 3), as shown. Homework Equations The field at position r⃗ due to the point charge is: E⃗ (r⃗...
  45. RedDelicious

    Use Gauss's Law to find the charge contained in the solid hemisphere

    Homework Statement edit: I had put this in the calculus section because it was a problem from Stewart but I guess it's closer to a physics problem considering the use of Gauss's Law. My apologies for any confusion this my [/B] I've been trying to do this problem without making use of the...
  46. Ryaners

    Electric field in two concentric conducting cylinders

    Homework Statement Two concentric cylindrical conducting shells of length L are separated by a vacuum. The inner shell has surface charge density +σ and radius ra. The outer shell has radius rb. Using Gauss’ Law, as a function of radius r find: The direction and magnitude of electric field...
  47. yeezyseason3

    Deriving Electric Field Inside Plastic Rod Using Gauss's Law

    Homework Statement A straight circular plastic cylinder of length L and radius R (where R ≪ L) is irradiated with a beam of protons so that there is a total excess charge Q distributed uniformly throughout the cylinder. Find the electric field inside the cylinder, a distance r from the center...
  48. K

    Gauss's Law Problem: long, cylindrical charge distribution

    Homework Statement Consider a long, cylindrical charge distribution of radius R with uniform charge density ρ. a) Using Gauss’s law, find the electric field at distance r from the axis, where r < R b) Using Gauss’s law, find the electric field at distance r from the axis, where r > R...
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