Electric Field/Electric Potential (Gradient Notation)

In summary, the correct notation for the electric field/electric potential gradient is \vec{E} = {-}\nabla{V(r)}, where \nabla represents the vector operator that takes a scalar field (such as the electric potential) and changes it to a vector field (such as the electric field) through partial differentiation with the addition of unit vectors (\hat{i}, \hat{j}, \hat{k}). The notation \vec{E}={-}\frac{\partial}{\partial{(x,y,z)}}{V(x,y,z)}{\hat{r}} is incorrect and \vec{E}={-}\nabla{V(r)} should be used instead.
  • #1
PFStudent
170
0

Homework Statement



Hey,

I have a question about Electric Field/Electric Potential gradient notation.

Since,

[tex]
{\vec{E}} = {-}{\nabla}{V(r)}
[/tex]

Which reduces to,

[tex]
\vec{E} = {-}{\nabla}{V(x, y, z)}
[/tex]

When expanded is,

[tex]
\vec{E} = {-}{\left[{\frac{\partial[V]}{\partial{x}}}{\hat{i}} + {\frac{\partial[V]}{\partial{y}}}{\hat{j}} + {\frac{\partial[V]}{\partial{z}}}{\hat{k}}\right]}
[/tex]

So using partial derivative notation can I write,

[tex]
{\vec{E}} = {-}{\vec{V}'_{xyz}}
[/tex]

So, is the above correct notation?

The reason I am hesitant is, because formally the gradient is defined as a vector operator that takes a scalar field (such as the electric potential) and changes it to a vector field (such as the electric field) through: partial differentiation with the addition of unit vectors ([tex]\hat{i}, \hat{j}, \hat{k}[/tex]).

However, writing it as below sort of implies the potential is a vector (which it isn't), but gives the impression that it is because of how the gradient is defined.

[tex]
{\vec{E}} = {-}{\vec{V}'_{xyz}}
[/tex]

So, is the above notation correct?

-PFStudent
 
Last edited:
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  • #2
PFStudent said:
However, writing it as below sort of implies the potential is a vector (which it isn't), but gives the impression that it is because of how the gradient is defined.

[tex]
{\vec{E}} = {-}{\vec{V}'_{xyz}}
[/tex]

So, is the above notation correct?

-PFStudent

I wouldn't use it. I would just leave it as:

[tex]
{\vec{E}} = {-}{\nabla}{V(r)}
[/tex]

Or

[tex]
{\vec{E}} = {-}{\nabla}{V}
[/tex]
 
  • #3
PFStudent said:
[tex]
{\vec{E}} = {-}{\vec{V}'_{xyz}}
[/tex]

So, is the above notation correct?

No, it is not correct. That is, there is no notation I know of that looks like that that is defined as the gradient of a scalar field.

As the above poster says, there is nothing wrong with [itex]\vec{E}=-\nabla V[/itex]
 
  • #4
Hey,

Yea, thanks for the input, I can see why that notation,

[tex]
{\vec{E}} = {-}{\vec{V}'{xyz}}
[/tex]

is wrong. Since, we are adding the components of a vector that is not the same as taking the partial derivative of a function with respect to each of the variables.

Since, all the gradient is doing is the following,

[tex]
\vec{E} = {-}{\nabla}{V(x, y, z)} = {-}{\left[{\frac{\partial}{\partial{x}}{\left[V\right]}} + {\frac{\partial}{\partial{y}}}{\left[V\right]}} + {\frac{\partial}{\partial{z}}}{\left[V\right]}}\right]}{\hat{r}}
[/tex]

Thanks,

-PFStudent
 
  • #5
Hey,

I've been thinking about this and I have a follow up question.

Since,

[tex]
\vec{E} = {-}{\nabla}{V(r)} = {-}{\left[{\frac{\partial}{\partial{x}}{\left[V\right]}} + {\frac{\partial}{\partial{y}}}{\left[V\right]}} + {\frac{\partial}{\partial{z}}}{\left[V\right]}}\right]}{\hat{r}}
[/tex]

and also,

[tex]
E = {-}{\frac{\partial}{\partial{r}}}{\left[{V(r)}\right]}
[/tex]

So then,

[tex]
\vec{E} = {-}{\nabla}{V(r)} = {-}{\frac{\partial}{\partial{r}}}{\left[{V(r)}\right]}{\hat{r}}
[/tex]

Now can I rewrite the above as below?

[tex]
\vec{E} = {-}{\frac{\partial}{\partial{(x, y, z)}}}{\left[{V(x, y, z)}\right]}{\hat{r}}
[/tex]

Which for [tex]{E}[/tex] can also be written as,

[tex]
{E} = {-}{\frac{\partial}{\partial{(x, y, z)}}}{\left[{V(x, y, z)}\right]}
[/tex]

So, is the notation for the above two equations correct?

Thanks,

-PFStudent
 

1. What is an electric field?

An electric field is a physical field that surrounds an electrically charged particle or object. It is a vector field, meaning it has both magnitude and direction, and is responsible for the movement and interaction of charged particles.

2. How is an electric field represented?

An electric field is typically represented by electric field lines, which show the direction and strength of the field. The density of the lines indicates the strength of the field, with closer lines representing a stronger field.

3. What is the difference between electric field and electric potential?

Electric field and electric potential are related but distinct concepts. Electric field is a physical field that exists around charged particles, while electric potential is the measure of the potential energy of a charged particle in an electric field. Electric potential is a scalar quantity, meaning it has magnitude but no direction.

4. What is gradient notation in regards to electric field and electric potential?

Gradient notation is a mathematical way of representing the strength and direction of an electric field or electric potential. It involves using the gradient operator (∇) to calculate the change in electric field or electric potential over a given distance. This notation is useful in complex systems where the electric field or potential varies in space.

5. How can I calculate the electric potential at a specific point in an electric field?

The electric potential at a specific point in an electric field can be calculated using the formula V = kQ/r, where V is the electric potential, k is the Coulomb's constant, Q is the charge creating the electric field, and r is the distance from the point to the charge. Alternatively, if the electric field is known, the potential can be calculated using the formula V = -∫E•dr, where E is the electric field and dr is a small distance element in the direction of the electric field.

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