Relationship between electric potential and electric field

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SUMMARY

The discussion focuses on the relationship between electric potential and electric field, specifically addressing how to identify scalar and vector fields. A scalar field associates a scalar value with each point in space, while a vector field, such as the electric field, is defined by its gradient. The participants emphasize that proving whether a function is a scalar or vector field can often be done through inspection rather than complex transformations. The key takeaway is that electric potential is a scalar field, while the electric field is a vector field derived from the gradient of the potential.

PREREQUISITES
  • Understanding of scalar and vector fields
  • Familiarity with electric potential and electric field concepts
  • Knowledge of gradients in vector calculus
  • Basic principles of electromagnetism
NEXT STEPS
  • Study the properties of scalar fields in physics
  • Learn about vector fields and their applications in electromagnetism
  • Explore the mathematical concept of gradients in vector calculus
  • Investigate the relationship between electric potential and electric field in more detail
USEFUL FOR

Students of physics, educators teaching electromagnetism, and anyone interested in understanding the fundamental concepts of electric fields and potentials.

Flotensia
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Homework Statement


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Homework Equations

The Attempt at a Solution


I could find how to solve #2,4, but I don't understand what #1,3 need to me. How can I prove some functions are scalar field or vector field?
 
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Flotensia said:
How can I prove some functions are scalar field or vector field?
The first step would be to find out what scalar and vector fields are. Once you do that, the rest should be fairly obvious.
 
Scalar field means the function of points associating scalar value. Is it clear?Then should I do rotational transformation to prove?
 
I think you're trying to say that a scalar fields associates a scalar with each point in space, which is correct. So you have to show (show, not prove) that the potential does just that. You shouldn't need to go to the trouble of doing any transformations.
 
Don't we have to show the quantity of point is scalar or vector??
 
You can tell that by inspection. There is nothing but scalars in #1. In #3 the result is obviously a vector, since it is a gradient, which is a vector by definition.
 

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