Using Gauss's law to calculate magnitude of electric field

In summary, you can use Gauss's law to calculate the electric field at a point within a sphere with a uniform or non-uniform charge distribution. You can also use Gauss's law to find the charge enclosed by a spherical surface.
  • #1
Signifier
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I understand how to use Gauss's law to calculate the electric field at some point for say a sphere with charge distributed uniformly in it. I am bit confused, though, about calculating the electric field at some point for a non-uniform charge distribution.

For example, say that I have a spherically symmetric negative charge distribution around the origin of my reference frame, with charge density p(r) given by -0.5e^(-r), where r is the distance from the origin (IE, the radius of a given sphere) (so, p(0) = -0.5 and p(1) ~ -0.2). If I wanted to calculate the magnitude of the electric field vector at some radius r, how would I do this?

Let's say I wrap up a portion of the charge distribution in a Gaussian sphere of radius r. The net flux through all tiles on this spherical Gaussian surface would be 4*pi*(r^2)*E, where E is the magnitude of the electric field at all points on the surface. Then, using Gauss's law and solving for E, I get

E = [k / (r^2)]Qenc(r)

Where Qenc(r) is the total charge enclosed by the closed surface at radius r. This is where I get stuck. Knowing the charge density function p(r) (in units C / m^3, let's say), how do I find the net enclosed charge by the sphere of radius r?

Any help would be appreciated.
 
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  • #2
Signifier said:
I understand how to use Gauss's law to calculate the electric field at some point for say a sphere with charge distributed uniformly in it. I am bit confused, though, about calculating the electric field at some point for a non-uniform charge distribution.

For example, say that I have a spherically symmetric negative charge distribution around the origin of my reference frame, with charge density p(r) given by -0.5e^(-r), where r is the distance from the origin (IE, the radius of a given sphere) (so, p(0) = -0.5 and p(1) ~ -0.2). If I wanted to calculate the magnitude of the electric field vector at some radius r, how would I do this?

Let's say I wrap up a portion of the charge distribution in a Gaussian sphere of radius r. The net flux through all tiles on this spherical Gaussian surface would be 4*pi*(r^2)*E, where E is the magnitude of the electric field at all points on the surface. Then, using Gauss's law and solving for E, I get

E = [k / (r^2)]Qenc(r)

Where Qenc(r) is the total charge enclosed by the closed surface at radius r. This is where I get stuck. Knowing the charge density function p(r) (in units C / m^3, let's say), how do I find the net enclosed charge by the sphere of radius r?

Any help would be appreciated.

By definition, the charge in a volume V is the integral of the volume charge density p over that volume, [itex] q_{enc}= \int dV \rho [/itex]. In your case, it is obviously easier to work in spherical coordinates and since the charge density is independent of the angle, you simply get [itex] q_{enc}= 4 \pi \int_0^R dr r^2 \rho(r) [/itex] where R is the location of the point where you want to evaluate the E field.

(I am assuming that this point is inside the charged sphere. If the point is
outside of the sphere then you obviously integrate up to the radius of the sphere only)

Patrick
 
  • #3
You'll have to integrate. Imagine thin spherical shells a distance [tex]r[/tex] from the origin of thickness [tex]dr[/tex].
 
  • #4
Okay, so I have a spherical shell of radius r and thickness dr. The volume enclosed by this shell would be the surface area of the sphere multiplied by the thickness (?), or 4*pi*(r^2)*dr. The charge enclosed by this would then be 4*pi*(r^2)*p(r)*dr. Then, to get the net charge enclosed by all of these shells from radius r = 0 to radius r = R, I integrate this expression from 0 to R as nrqed has.

Thanks durt and nrqed, I get it now!
 

1. How do you use Gauss's law to calculate the magnitude of electric field?

To use Gauss's law, you need to choose a closed surface around the charge or charges you want to calculate the electric field for. Then, calculate the flux through this surface by multiplying the electric field at each point on the surface by the area element and taking the sum. Finally, divide the flux by the permittivity of free space to get the magnitude of the electric field.

2. What is the significance of using Gauss's law to calculate the magnitude of electric field?

Using Gauss's law allows us to calculate the electric field at a point without having to consider every single charge in the system. This makes calculations easier and more efficient, especially for systems with multiple charges.

3. How does the shape of the closed surface affect the calculation using Gauss's law?

The shape of the closed surface does not affect the calculation as long as it encloses the charges of interest. However, choosing a convenient shape can simplify the calculation and make it easier to determine the electric field at the point of interest.

4. Can Gauss's law be used for any type of charge distribution?

Yes, Gauss's law can be used for any type of charge distribution, as long as the distribution is spherically symmetric. In cases where the distribution is not spherically symmetric, the law can still be applied, but the calculation may be more complex.

5. How can Gauss's law be extended to calculate the electric field for non-spherically symmetric charge distributions?

To calculate the electric field for non-spherically symmetric charge distributions, we can use the general form of Gauss's law, which involves the divergence of the electric field. This allows us to integrate the electric field over a volume, rather than just a surface, and take into account the non-spherical nature of the charge distribution.

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