# Electric field with electric potential

• Kernul
In summary: Then use that ##E=\frac{λ}{2πε_0r}##.In summary, the problem involves a metallic hollow cylinder with a diameter of 4.2 cm and a wire with a diameter of 2.68 μm running along its axis. The voltage between the cylinder and wire is 855 V. The goal is to find the electric field at the wire surface and the cylinder surface. Using the equation for electric potential, the radius of the wire and cylinder were calculated and the first equation was used to find the electric field. The concept of boundary conditions and the use of cylindrical-polar coordinates were discussed. Applying Gauss' Law, it was determined that the electric field is perpendicular to the metallic surfaces and has

## Homework Statement

A metallic hollow cylinder has a diameter of ##4.2 cm##. Along his axis there is a wire having a diameter of ##2.68 \mu m##(considering it as a hollow cylinder). Between the cylinder and the wire there is a voltage of ##855 V##.
What is the electric field on the wire surface and the cylinder surface?

## Homework Equations

Electric potential:
##\int_{A}^{B} \vec E_0 \cdot d\vec l = V_0(A) - V_0(B)##
For a generic point ##P##
##V_0(x, y, z) = \int_A^P \vec E_0 \cdot d\vec l + V_0(A)##

## The Attempt at a Solution

First I got the radius from the diameters, so:
##D_1 = 2.68\mu m = 2.68 * 10^-6 m##
##D_2 = 4.2 cm = 4.2 * 10^-2 m##
##R_1 = 1.34 * 10^-6 m##
##R_2 = 2.1 * 10^-2 m##

At this point I know that ##V_0(R_1) - V_0(R_2) = 855 V##, so I have to use the first equation to find out ##E_0##. The problem is that I don't understand what ##E_0## I am calculating with this equation:
$$\int_{R_1}^{R_2} \vec E_0 \cdot d\vec l = V_0(R_1) - V_0(R_2)$$
Is it the wire surface? Or am I calculating the electric field inside the cylinder but outside the wire?

You could try the differential form: ##\vec E=-\vec \nabla V## and select a suitable coordinate system and boundary conditions.

Boundary conditions? What do you mean by that?

Kernul said:

## Homework Statement

A metallic hollow cylinder has a diameter of ##4.2 cm##. Along his axis there is a wire having a diameter of ##2.68 \mu m##(considering it as a hollow cylinder). Between the cylinder and the wire there is a voltage of ##855 V##.
What is the electric field on the wire surface and the cylinder surface?

## Homework Equations

Electric potential:
##\int_{A}^{B} \vec E_0 \cdot d\vec l = V_0(A) - V_0(B)##
For a generic point ##P##
##V_0(x, y, z) = \int_A^P \vec E_0 \cdot d\vec l + V_0(A)##At this point I know that ##V_0(R_1) - V_0(R_2) = 855 V##, so I have to use the first equation to find out ##E_0##. The problem is that I don't understand what ##E_0## I am calculating with this equation:
Where is that equation from? You should use only such formulas which are explained.
You certainly know that the line integral between two points does not depend on the path taken and ##\int_{A}^{B} \vec E \cdot d\vec l = V(A) - V(B)## where the integral goes along the radius now, from the wire surface to the surface of the cylinder. You have to know how the electric field depends on the radius. For that, use Gauss' Law.

Kernul said:
Boundary conditions? What do you mean by that?
Boundary conditions are the conditions on the boundary of the problem. Your problem has two boundaries - which is the cylinder surfaces.
The relevant condition on those boundaries is the potentials there.
Do you not know how to solve differential equations?

Do you know how to use cylindrical-polar coordinates?
Do you know Gauss' Law?

Differential equations are those where you derive for each unknown and take the others as constant. (x, y, z in this case, since we have three coordinates)
I never used the cylindrical and polar coordinates but I read about them in the book.
Yes, I know the Gauss' Theorem.

ehild said:
You have to know how the electric field depends on the radius. For that, use Gauss' Law.
But how? I don't know the charge distribution.

Kernul said:
Differential equations are those where you derive for each unknown and take the others as constant. (x, y, z in this case, since we have three coordinates)
I never used the cylindrical and polar coordinates but I read about them in the book.
Yes, I know the Gauss' Theorem.But how? I don't know the charge distribution.
But you know the symmetry of the problem. You can assume that the charge distribution is even, the charge is λ on unit length of the wire.
And you know the potential difference between the metallic surfaces. The metallic surfaces are equipotentials, and the electric field is perpendicular to them, and is the same in every direction. The blue lines show the electric field lines, and the grin circle is the cross section of a Gaussian cylinder. Apply Gauss Law to get the electric field intensity at distance r from the axis.

## 1. What is an electric field?

An electric field is a region in space surrounding a charged object where other charged objects will experience a force. This force is either attractive or repulsive depending on the charges of the objects involved.

## 2. How is an electric field created?

An electric field is created by a charged object due to the presence of its electric charge. The strength of the electric field is determined by the magnitude of the charge and the distance from the charged object.

## 3. What is electric potential?

Electric potential is the amount of electric potential energy per unit charge at a given point in an electric field. It is a measure of the work required to move a unit charge from one point to another in the electric field. The unit of electric potential is volts (V).

## 4. How is electric potential related to electric field?

The electric field and electric potential are related by the equation E = -∇V, where E is the electric field, ∇ is the gradient operator, and V is the electric potential. This means that the electric field is the negative gradient of the electric potential.

## 5. What is the difference between electric potential and electric potential energy?

Electric potential is a measure of the electric potential energy per unit charge at a given point in an electric field. Electric potential energy is the amount of work required to move a charge from one point to another in an electric field. They are related by the equation PE = qV, where PE is the electric potential energy, q is the charge, and V is the electric potential.