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astro2cosmos
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if E=-curl of V, E is vector quantity(3 components) & V is scalar quantity (1 component) then how can one function possibly contain all the information that 3 independent function carry?
The first thing to point out is that a scalar isn't simply a one-dimensional vector, so it itsn't really correct to say a scalar has one component. Secondly, the curl operator on acts on vector fields. Finally, it is very easy for a scalar field to contain all the information necessary to generate a vector field. For example, the gradient of a scalar field is defined as,astro2cosmos said:if E=-curl of V, E is vector quantity(3 components) & V is scalar quantity (1 component) then how can one function possibly contain all the information that 3 independent function carry?
Hootenanny said:The first thing to point out is that a scalar isn't simply a one-dimensional vector, so it itsn't really correct to say a scalar has one component. Secondly, the curl operator on acts on vector fields. Finally, it is very easy for a scalar field to contain all the information necessary to generate a vector field. For example, the gradient of a scalar field is defined as,
[tex]\nabla V\left(x,y,z\right) = \left(\frac{\partial V}{\partial x},\;\frac{\partial V}{\partial y},\; \frac{\partial V}{\partial z}\right)[/tex]
Hence, the gradient of a scalar field results in a vector which represents how fast the scalar field V is changing in the directions parallel to the three axes.
Do you see?
Edit: I see that I was beaten to it.
In what sense? What equation are you using to determine the electric field?astro2cosmos said:but is it right to say that the 3 components of electric field are independent??
In this equation, the field refers to the electric field, which is a vector quantity that describes the direction and magnitude of the force that a charged particle experiences. The potential refers to the electric potential, which is a scalar quantity that describes the amount of potential energy per unit charge at a given point in space.
In this equation, the electric field is related to the electric potential through the curl operator, which is a mathematical operation that describes how a vector field changes in space. The electric field can be found by taking the negative curl of the electric potential.
E=-curl of V is a fundamental equation in electromagnetism that describes the relationship between the electric field and electric potential. It is used to calculate the electric field in a given region of space, which is important for understanding the behavior of charged particles and the flow of electricity.
Yes, E=-curl of V can be applied to other types of fields, not just electric fields. This equation is a general form of the Maxwell-Faraday equation, which is one of the four Maxwell's equations that govern the behavior of electric and magnetic fields in space. It can be used to calculate the magnetic field from the magnetic potential as well.
E=-curl of V is used in a variety of practical applications, such as in electrical engineering, telecommunications, and physics research. It is essential for understanding and designing electrical circuits and devices, as well as for studying electromagnetic phenomena and developing new technologies.