Calculation of electric field from a set of equipotential surfaces

In summary, the problem is to find the direction and value of the electric field in a region with concentric hemispherical surfaces, each with a different potential value and spacing. The electric field will be in the radial direction and can be calculated using the negative gradient of potential and a Gaussian surface with no charge inside. The field decreases with 1/r^2 and can be solved for using a radial line integral.
  • #1
kihr
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Homework Statement


A set of concentric hemispherical surfaces is given, each of which is an equipotential surface. These concentric surfaces do not, however, have the same value of potential, and the potential difference between any two surfaces is also not constant. The surfaces are spaced apart at the same radial distance between each other. I need to find the direction and value of the electric field in this region.


Homework Equations


The electric field will be in the radial direction as the field has to be normal to the equipotential surface at each point. Its direction will be along the negative gradient of the potential (E = - dV/dr).


The Attempt at a Solution



I need some clues as to how to calculate the value of the electric field. This would have been easy if the relation between V and r were known. I guess this needs to be derived. Please give me some hints as to how to tackle this. Thanks.
 
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  • #2
take a guassian surface with no charge inside to convince yourself the field decreases with 1/r^2, between any 2 plates

then take a radial line integral of the field, which must be equal to the potential difference to solve for E
 
  • #3
With your tips I was able to solve the problem. Many thanks.
 

1. How is the electric field calculated from a set of equipotential surfaces?

The electric field is calculated by finding the gradient of the potential function at each point on the set of equipotential surfaces. This is done by taking the partial derivatives of the potential function with respect to the x, y, and z coordinates.

2. What is an equipotential surface?

An equipotential surface is a surface where the potential function is constant at every point. In other words, the electric potential is the same at every point on the surface.

3. What is the significance of calculating the electric field from a set of equipotential surfaces?

Calculating the electric field from a set of equipotential surfaces allows us to understand the behavior of electric fields in a given region. It also helps us visualize the electric field lines and determine the direction and magnitude of the electric field at different points.

4. Can the electric field be calculated at any point in the region from a set of equipotential surfaces?

Yes, the electric field can be calculated at any point in the region using the gradient of the potential function at that point. This is because the electric field is perpendicular to the equipotential surfaces and points in the direction of decreasing potential.

5. How does the distance between equipotential surfaces affect the electric field?

The closer the equipotential surfaces are, the stronger the electric field will be between them. This is because the potential difference between the surfaces is smaller, resulting in a larger gradient and thus a stronger electric field. Conversely, a larger distance between equipotential surfaces means a weaker electric field between them.

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