Space Dependence of Electric potential

In summary: So the line of charge would be in the \hat{k} direction and it would be pointing towards the positive x-axis.
  • #1
PhysicsRob
8
0

Homework Statement



The space dependence of an electric potential V([itex]\vec{r}[/itex]) = V(x,y,z)=V0ln((sqrt{x2 + y2})/a)

1. What is the electric field at position [itex]\vec{r}[/itex] = <x,y,z>?

2. Explain how the electric field looks in general. Make a sketch.

3. What object would produce an electric field like this?

The Attempt at a Solution



I haven't really tried much yet. What really confuses me about this question is the part in the prompt that comes after the equals sign. I'm not really sure what I'm supposed to do with it. To find the electric field do they just want me to take the gradient of the potential or something? That doesn't seem right because there's only a constant V0. I'm a bit lost...
 
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  • #2
PhysicsRob said:

Homework Statement



The space dependence of an electric potential V([itex]\vec{r}[/itex]) = V(x,y,z)=V0ln((sqrt{x2 + y2})/a)

1. What is the electric field at position [itex]\vec{r}[/itex] = <x,y,z>?

2. Explain how the electric field looks in general. Make a sketch.

3. What object would produce an electric field like this?

The Attempt at a Solution



I haven't really tried much yet. What really confuses me about this question is the part in the prompt that comes after the equals sign. I'm not really sure what I'm supposed to do with it. To find the electric field do they just want me to take the gradient of the potential or something? That doesn't seem right because there's only a constant V0. I'm a bit lost...

Yes, the electric field is the (negative) gradient of the electric potential. What do you mean "in the prompt?" In any case, V is certainly not constant. It varies with x and y.
 
  • #3
Well I understand the idea that V([itex]\vec{r}[/itex]) varies with x and y, but isn't V0 a constant? That's what's confusing me. Generally when things have the subscript "0", they're looked at as a constant. If this is true, then how would I take the gradient of the voltage?

And when I say "In the prompt", I mean the information they give you before they ask the questions, aka: "The space dependence of an electric potential V(r⃗ ) = V(x,y,z)=V0ln((sqrt{x2 + y2})/a)"
 
  • #4
Uhh...yes V0 is a constant, but it is being multiplied by something that is a function of x and y. The V(x,y,z) that you have been given in this problem is not constant.
 
  • #5
Ohhhh, wait a second. I see what I'm doing now.

So by doing the gradient I got that:
Ex = -(V0x)/(x2 + y2)
Ey = -(V0y)/(x2 + y2)
Ez = 0

And if you model sqrt{x2 + y2} as a vector "r", you get that the field has an inverse dependence on r and that the object producing the field would probably be a line of charge, correct?
 
  • #6
PhysicsRob said:
Ohhhh, wait a second. I see what I'm doing now.

So by doing the gradient I got that:
Ex = -(V0x)/(x2 + y2)
Ey = -(V0y)/(x2 + y2)
Ez = 0

And if you model sqrt{x2 + y2} as a vector "r", you get that the field has an inverse dependence on r and that the object producing the field would probably be a line of charge, correct?

Well, sqrt(x^2+y^2) is the scalar "r", but yes, I'd say it looks like a line of charge. Can you say where the line of charge is and what direction it's pointing?
 
  • #7
Well... if you were to move along the z-axis while keeping the same x and y coordinates, your change in voltage would be 0 since the voltage doesn't depend on z. So the line of charge would be along the z axis and just points in the [itex]\hat{k}[/itex] direction, right?
 
  • #8
PhysicsRob said:
Well... if you were to move along the z-axis while keeping the same x and y coordinates, your change in voltage would be 0 since the voltage doesn't depend on z. So the line of charge would be along the z axis and just points in the [itex]\hat{k}[/itex] direction, right?

Right!
 

1. What is the Space Dependence of Electric Potential?

The space dependence of electric potential refers to how the electric potential changes as the distance from a source charge or charges varies. This is a fundamental concept in electromagnetism and is used to understand the behavior of electric fields and charges.

2. How does the Space Dependence of Electric Potential affect the movement of charges?

The space dependence of electric potential determines the direction and magnitude of the electric field, which in turn affects the movement of charges. Charges will naturally move from areas of high potential to areas of low potential, following the path of the electric field lines.

3. What is the mathematical expression for the Space Dependence of Electric Potential?

The mathematical expression for the space dependence of electric potential is given by the equation V = kQ/r, where V is the electric potential, k is the Coulomb constant, Q is the source charge, and r is the distance from the source charge.

4. How is the Space Dependence of Electric Potential related to the concept of electric potential energy?

The space dependence of electric potential is directly related to electric potential energy. The electric potential at a point is equal to the electric potential energy per unit charge at that point. This means that the space dependence of electric potential can be used to calculate the potential energy of a charge at a specific point in space.

5. How can the Space Dependence of Electric Potential be measured?

The space dependence of electric potential can be measured using a voltmeter. By placing the voltmeter at different distances from a source charge and recording the potential difference, the relationship between distance and potential can be determined. This information can then be used to create a graph and analyze the space dependence of electric potential.

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