Field vs Potential: E=-curl of V

[astro2cosmos]

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?

 

[Born2bwire]

You have your operators mixed up. Curl only operates on other vectors and produces a vector. The relationship between the electric field and electric potential is through the gradient operator. The gradient operator acts on a scalar and produces a vector. Imagine a topographical map. A topographical map is a 2D plane that maps the vertical height of the landscape. The vertical height is shown by lines of iso-height (yeah, that's the wrong word but I mean lines that show a continuous line of constant height). The gradient operator, acting on the height, would produce vectors that are normal to these lines of iso-height, they point in the direction of greatest change in the scalar as a function of the coordinate axes.

So if our scalar is dependent upon three variables, say the three spatial coordinates, then the gradient would produce a vector that points in the direction of greatest change in the coordinate space.

 

[Hootenanny]

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,

$$\nabla V\left(x,y,z\right) = \left(\frac{\partial V}{\partial x},\;\frac{\partial V}{\partial y},\; \frac{\partial V}{\partial z}\right)$$

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.

 

[astro2cosmos]

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,

$$\nabla V\left(x,y,z\right) = \left(\frac{\partial V}{\partial x},\;\frac{\partial V}{\partial y},\; \frac{\partial V}{\partial z}\right)$$

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.

yes you may right, but is it right to say that the 3 components of electric field are independent??

 

[Hootenanny]

but is it right to say that the 3 components of electric field are independent??

In what sense? What equation are you using to determine the electric field?

 

[Born2bwire]

Independent in what way? As vector components they are orthogonally independent. As functions, the magnitude of a vector component can be dependent upon the other spatial coordinates. That is, the x-component of the electric field can still be dependent or indepedent of the x, y, and/or z coordinates.

EDIT: Curses! Hootenanny wins this round.