Derivation of Electric Field with Gauss's Law

In summary, the conversation discusses the calculation of the magnitude of the electric field E(r) at a distance r>rb from a solid ball with uniform charge density ρ. The equation is derived using Gauss's law and involves dividing the volume of the ball by the electric constant ε0. The book answer differs from the derived answer by a constant of 1/3, which may be due to a difference in the details of the lefthand side of the equation.
  • #1
k_squared
64
0
I did everything I could to solve the following problem:
A solid ball of radius rb has a uniform charge density ρ.

What is the magnitude of the electric field E(r) at a distance r>rb from the center of the ball?
E(r) =

My third attempt went like this: qencl=[ρ(4/3)(π)rb3]

EV=[ρ(4/3)(π)rb3]/(ε0)
E(4/3)πr3=[ρ(4/3)(π)rb3]/(ε0)

And ah, well, a little simple division and cancelling leads to:
[ρrb3/[ε0r3]

However, the book answer is 1/3 my answer. Could someone please tell me where this constant develops?
 
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  • #2
Try Gauss's law. Start by writing the equation for Gauss's law.
 
  • #3
k_squared said:
EV=[ρ(4/3)(π)rb3]/(ε0)
The righthand side is okay. Check the details on the lefthand side of this equation.
 
  • #4
Why did you multiply the electric field with the volume?
 

FAQ: Derivation of Electric Field with Gauss's Law

How is Gauss's Law used to derive the electric field?

Gauss's Law is a fundamental law in electrostatics that relates the electric flux through a closed surface to the charge enclosed within that surface. By applying this law to a specific situation, such as a point charge or a uniformly charged sphere, we can derive an equation for the electric field at any point in space.

What is the mathematical expression for Gauss's Law?

The mathematical expression for Gauss's Law is ∮E • dA = Qenc / ε0, where ∮E • dA represents the electric flux through a closed surface, Qenc is the charge enclosed within that surface, and ε0 is the permittivity of free space.

How do you determine the direction of the electric field using Gauss's Law?

The direction of the electric field can be determined by using the right-hand rule. If the enclosed charge is positive, the electric field lines will point away from the charge, and if the enclosed charge is negative, the electric field lines will point towards the charge.

Can Gauss's Law be used for any shape or distribution of charge?

Yes, Gauss's Law can be used for any shape or distribution of charge, as long as the charge is enclosed within the closed surface being considered. However, the calculations become more complex for non-symmetric charge distributions.

What are the units of the electric field derived from Gauss's Law?

The units of the electric field derived from Gauss's Law are newtons per coulomb (N/C) or volts per meter (V/m). These units represent the force per unit charge or the potential gradient, respectively.

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