Apparent disagreement between Coulomb's Law and Gauss' Law

In summary, the conversation discusses two different methods for calculating the electric field at a point between two charges, using Coulomb's Law and Gauss's Law. While Coulomb's Law yields an electric field of (1/2πε0)Q/r2, Gauss's Law gives a different result of (1/4πε0)Q/r2 due to the total field not being constant across the Gaussian surface. It is clarified that the error lies in assuming symmetry of the electric field, and that the integral must be done to accurately calculate the electric field.
  • #1
shj
4
1
pub?w=617&h=598.png

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 includes point P is drawn above.

Using Coulomb's Law, we can find the electric field at point P:
E=2*((1/4πε0)Q/r2)=(1/2πε0)Q/r2)
(since the electric field vectors caused by both charges have the same magnitude and add at point P)

However, if I try to use Gauss's Law to calculate the electric field at point P, I get:
E*4πr2=Qenclosed/ε0, or
E=(1/4πε0)Q/r2)
(since the Gaussian surface is symmetric to the electric field, I simplified the surface integral to E*4πr2)

The two calculations differ! Can someone please clarify the error?!
 

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  • #2
Simple. The total field is not constant across the Gauss surface. You would really need to do the integral.

In other words, here is your error:
shj said:
(since the Gaussian surface is symmetric to the electric field, I simplified the surface integral to E*4πr2)
Not symmetric for the total field!

Note: Welcome to PF!
 
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  • #3
Stavros Kiri said:
Simple. The total field is not constant across the Gauss surface. You would really need to do the integral.

In other words, here is your error:

Not symmetric for the total field!

Note: Welcome to PF!
Oh alright. Thank you.
 
  • #4
shj said:
Oh alright. Thank you.
You're welcome!
 
  • #5
shj said:
since the Gaussian surface is symmetric to the electric field,
The electric field is not spherically symmetric

Edit: oops, I am too late. Good job @Stavros Kiri
 
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1. What is Coulomb's Law and Gauss' Law?

Coulomb's Law and Gauss' Law are two fundamental laws in electromagnetism. Coulomb's Law states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Gauss' Law, on the other hand, states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space.

2. What is the apparent disagreement between Coulomb's Law and Gauss' Law?

The apparent disagreement between Coulomb's Law and Gauss' Law arises when we consider situations where there is a non-uniform distribution of charge. According to Coulomb's Law, the force between two charged particles is directly proportional to the distance between them. However, Gauss' Law does not take into account the distance between the charges, and instead, only considers the total charge enclosed by a surface.

3. How do we resolve this apparent disagreement?

The apparent disagreement between Coulomb's Law and Gauss' Law can be resolved by considering the concept of charge density. Charge density is the amount of charge per unit volume. By taking into account the charge density, we can use Gauss' Law to calculate the total charge enclosed by a surface, and then use Coulomb's Law to calculate the force between two charged particles at a specific distance.

4. Are there any real-life examples that demonstrate the relationship between Coulomb's Law and Gauss' Law?

Yes, there are many real-life examples that demonstrate the relationship between Coulomb's Law and Gauss' Law. One example is a charged spherical shell. According to Gauss' Law, the electric field inside a charged spherical shell is zero, as there is no net charge enclosed by the surface. However, according to Coulomb's Law, the force between two charges on the surface of the shell will still be non-zero if they are not directly opposite each other.

5. How do Coulomb's Law and Gauss' Law contribute to our understanding of electromagnetism?

Coulomb's Law and Gauss' Law are fundamental laws that help us understand the behavior of electric charges and their interactions. They play a crucial role in many areas of physics, such as electronics, electrostatics, and electrodynamics. Together, these laws provide a comprehensive understanding of how electric charges behave and how they influence the world around us.

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