What is Gauss' law: Definition and 339 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.
Homework Statement
Gauss' Law states:
∫∫ E.dS = ∫∫∫ div(E) dV = Qenc/ε₀
The proof is as follows (this is from Marsden's Vector Calculus 5e):
Let M be a elementary region in ℝ3. Then if (0,0,0) ∉ ∂M, we have:
∫∫∂M r.n/r3 dS
= 4π if (0,0,0) ∈ M
= 0 if (0,0,0) ∉ M
Construct a sphere of...
Homework Statement
Using Gauss' law, derive the expression for the electric field intensity vector of an infinite sheet of charge in free space.
Homework Equations
The Attempt at a Solution
See figure attached for their solution and the figure that goes with the problem.
What...
In general I just wanted to see if I was setting this problem up correctly.
We have a cube centered around the origin and a point charge at the origin. The task is to find the flux & confirm Gauss' Law. We are however to complete the integral ourselves. So imagining the top of the cube...
Homework Statement
The total electric flux from a cubical box 40.0 cm on a side is 4700 Nm^2/C. What charge is enclosed by the box?
Homework Equations
Gauss Law: surface integral of E-field-vector * dA-vector = Q_enclosed/epsilon naught
The Attempt at a Solution
The reason I am...
1. As a member of a team of storm physicists, you are attempting to replicate lightning by chargeing two long cables stretched over a canyon. one cable will attain a highly positive and uniform density of -\lambda and the other will attain the same amount of charge density, but opposite in...
Homework Statement
A point charge, q, is at the center of an insulating hollow sphere with inside radius, a, and outside radius, b. The spherical shell has charge per unit volume, rho. You may assume that q and rho are positive.
Using Gauss' law, find the magnitude of the electric field...
Homework Statement
The Electric field E produced by an unknown charge distribution p (rho) is E(r)= (constant)*((exp(-ar))/r^2)*(r_hat).
a.) Use Gauss' law in differential for to determine p(rho)
b.) Find the total charge q_tot by directly integrating p(rho), and show that it is 0.
c.)...
I'm having some difficulty applying Gauss's Law to metal plates. From what I've studied it seems to me that Gauss's Law only really works when there is spherical symmetry, in all other cases it is an approximation. Am I correct in inferring this?
However, in a case where the distance between...
Gauss' Law-Can't we find the Electric Field (In the vector form) from Gauss' Law? Because in most of the problems I have been doing like the case of a Charge in a solid sphere, I can find the Magnitude of Electric Field by Gauss' Law but not the Electric Field. Am I wrong here?
Homework Statement
how do I know which equation to use for electric field? E=a/2e0 or E=a/e0 when a = surface charge density.
Homework Equations
The Attempt at a Solution
Homework Statement
[PLAIN]http://smartphysics.com/images/content/EM/03/h3_lineF.png
A cylinder of radius a = 6.1 cm and height h = 9.7 cm is aligned with its axis along the y-axis. An infinite line of charge is placed at x=3.05cm where λ1 = -2 μC/cm and λ2 = 6 μC/cm at x=-3.05 cm. Point P is...
The first Maxwell equation, Gauss' law makes a clear distinction between "inside" and "outside". But such a distinction can't be made in a compactified space (e.g. circle). Does that mean that the universe isn't compactified in a sense that if one was to move in a "straight" line one would never...
Homework Statement
A hollow sherical conducting shell is suspended in air by an insulated string. The total charge on the conductor is -6 microCoulombs. If an additional point charge of +2 microCoulombs is placed in the hollow region inside the shell what is the total charge induced on the...
I'm stumped on a problem about Gauss' Law for Magnetism because I don't know what "local form" is. From context, I suspect that local form refers to the expression of an equation with the del operator (i.e. del x E = (charge density/(E0)), but I'm not sure. I've been self-teaching with Tipler...
Homework Statement
This isn't about a specific problem, but it is based off of a homework problem. There is an insulating sphere (from radius 0 to a), and it is concentric with a spherical conducting shell (from radius b to c). If I know the charge of the insulating sphere and the net charge...
Homework Statement
A high voltage coaxial cable is used to supply power to an x-ray generator. The cable consists of an inner wire of radius r=1 mm and a thin hollow outer conductor of radius R= 10 mm. The inner wire and the outer shield have the same charge density per unit length of ...
Homework Statement
Charge is uniformly distributed along the x-axis with density ß. Use Gauss' Law to find the electric field it produces, and use this to calculate the work done on a charge Q that moves along the y-axis from y = a to y = b.
Homework Equations
Gauss' Law: e0 * electric...
Homework Statement
Trying to find E field inside a sphere radius a with volume charge density rho = k/r
Homework Equations
Gauss' law
The Attempt at a Solution
I set up a spherical shell radius R (R<a)
I found the charge inside by integrating rho from 0 to R (Q = 2*pi*a*R^2)...
Homework Statement
Imagine a solid, annular sphere. At the center of the hollow is a point charge +Q. The inner radius of the sphere is r0, and the outer radius is R. Assume the charge density p = p0/r (for r0 < r < R). Calculate using the integral form of Gauss's Law the electric field in all...
Homework Statement
show that the E outside an infinitely long rod of radius R with a uniform charge density p is E = pR^2/2r(e_0)
Homework Equations
gauss' law EA=q/e_0
The Attempt at a Solution
I know how to solve this and get the correct answer but I don't totally understand it...
Homework Statement
A long, solid, non-conducting cylinder of radius 8 cm has a non-uniform volume density, ρ, that is a function of the radial distance r from the axis of the cylinder. ρ = A*r2 where A is a constant of value 2.9 μC/m5.
What is the magnitude of the electric field 7 cm...
Homework Statement
A uniformly charged ball of radius a and charge -Q is at the center of a hollow metal shell with inner raduis b and outer radius c. \The hollow sphere has net charge +2Q.
Determine the Electric Field Strength at r when r is,
r < a
a < r < b
b< r < c
r > c...
Hey there, just had a question about Gauss' law, should be relativity simple however the explanation we were given was quite poor and only seems to apply well to the examples we were given. (This isn't homework).
I (think I ) know the equation for Gauss' Law and what it means, that basically...
Homework Statement
When I try working out the example below from PlanetPhysics, I wind up with 2PI rather than 4PI in my answer. Should I be considering my result valid for only a hemisphere and double it for a sphere--or am I just making a mistake in my math?
"As an example of the...
2 questions:
1. A point charge of 1.84 microC is at the center of a cubical Gaussian surface 55cm on edge. Find \Phi_E through the surface.
So here I was thinking, well the shape doesn't matter so the surface can be a sphere, so I calculated it for a sphere and it was correct (taking the...
I've seen many examples of spheres, cylinders, and planes and I'm trying to understand when to use r or R in the equation E A = 1/4piEo * Qenc. I've seen examples where the r's will cancel giving a simplified answer and then others where you have something like R^2 / r^2.
So my question is...
Homework Statement
A solid sphere of radius R carries a volume charge density \rho = \rho_0e^{r/R}, where \rho_0 is a constant and r is the distance from the center.
Find an expression for the electric field strength at the sphere's surface.
Homework Equations
\int\vec{E}.d\vec{A} =...
I have a few questions related to finding the electric field of an object.
1. What's the difference between a conducting object (sphere, cylinder) vs. a non conducting object? Is the charge inside a conducting and nonconducting sphere both zero if the surface charge density is uniform? What...
Homework Statement
A hollow sphere of radius r_1 is placed at the centre of a larger hollow sphere of radius r_2.
Both spheres have a uniformly distributed total charge of +q
Find the preassure p(r_2 , q) which acts on the outer sphere.
Homework Equations
\oint\textbf{E.n}dS = 4\pi k Q
p =...
Before I get into the question I'd just like to state that this is not homework, but questions in my book that I'm going through to prepare myself for the midterm in one week. I got stuck at a few questions, here's the first one. I won't ask the next until I'm done with this and so forth...
The differential form of Gauss' law states that
\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}.
So the divergence of the electric field is the chargedensity divided by epsilon zero.
I just wondered.. since divergence is a local or "point" property. Is the chargedensity in this law also...
1. An infinitely long cylinder of radius R contains a uniform charge density Rho. Calculate the electric field using Gauss' law for r> R and R>r
3. I attached a pdf with my attempt at r>R. My answer doesn't agree with those given...
I have a 5mm diameter, 10mm long magnet, a short (6mm) coil of 50mm diameter, and an oscilloscope. When I move the magnet through the coil, the oscilloscope shows that the signal (voltage) looks pretty much like the derivative of a Gaussian function. So there must be magnetic flux maximum when...
According to various EM texts (Feynman, Griffiths, ...) Gauss’ law holds only in electrostatic situations. But using the point charge electric field solutions, I have found to date that it holds for a relativistically oscillating charge (wA = .99c) within at least 3 flux integration surfaces...
Hello all. I have been trying to learn some Physics in my spare time and I came across Gauss' Law. I've been thinking about different cases and conditions and I have been confused by the actual meaning of Electric flux being 0.
Homework Statement
If I consider a sheet of uniform charge per...
I have been studying Gauss' law and almost all of the problems I have been doing just have me integrate dA alone into A. I was wondering when do you actually have to do some more in depth integration.
Homework Statement
http://img685.imageshack.us/img685/9501/10953060.png"
Homework Equations
The Attempt at a Solution
I know you use Gauss' Law, but why wouldn't the charges outside of the shell induce a charge on the shell, which would then affect the field at P?
Pretty much worked out, but stuck! Gauss' Law problem
Homework Statement
"Consider a charge density distribution in space given by \rho = \rho_0 e^{-r/a}, where \rho_0 and a are constants. Using Gauss' Law, derive an expression for the electric field as a function of radial distance, r...
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 a total charge of -1q and the outer shell has a total charge of +4q...
Homework Statement
A non-conducting, infinitely long, cylindrical sheath has inner radius r=10 m, outer radius r=15 m and a uniform charge density of 9 nC/m^3 spread throughout the sheath. Magnitude of electric field at r=5, r=12, r=17?
Homework Equations
Q=rho(Volume) and phi=EA...
This is a general question, I was reading my textbook and this statement confuses me:
"The symmetry of the electric field must match the symmetry of the charge distribution"
this is said regarding symmetry in relation to GAuss' law.
I do not understand what they mean by symmetry of charge...
I have a cube with sides l and therefore a total volume of l^3. The E-field is ax + by + cz where a, b, and c are known constants. It says to find the total E-field. The answer is (a + b + c)*l^3. I have no idea how they get this. I tried summing up all of the sides separately bit I kept getting...
Homework Statement
For the given flux density: \vec{D}=(2y2+z)\widehat{x}+(4xy)\widehat{y}^+(xz)\widehat{z}
a)Determine the charge density.
b)Find the total charge enclosed if the surface is 0<x<1, 0<y<1, 0<z<1 (unit cube)
c)Confirm Gauss’s law by finding the net flux through the surface of...
If I have a hypothetical infinite 3 dimensional universe uniformly charged I can argue that the electric field is zero at any given point in space by symmetry. However, if I invoke Gauss' Law I find a non-zero flux of electric field through a gaussian surface encompassing any amount of charge...
Homework Statement
A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by
ρ(r) = 3αr/(2R) for r is less than or equal to R/2
ρ(r) = α[1- (r/R)^2] for R/2 is less than or qual to r which is less than or...
I don't really know much about serious business electrostatics. I've only taken the AP Physics C E&M exam (using the integral form), but I was looking at wikipedia and I was curious about the differential form of Gauss' Law.
I don't understand what I'm doing wrong with this. I'm trying to...
Homework Statement
I actually just have two questions about the law itself.
1.) Is Gauss's Law just the projection of a shape onto a perpindicular surface to the E field,times the E field?
I ask this because there is an example in my book which shows a 3D incline with a horizontal E...
Homework Statement
A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find:
(a) the charge density of each face of the plate and
(b) the total charge on each face
Homework...