Finding a vector field perpendicular to the surface of a sphere

In summary, the conversation discusses the process of determining if a given vector field is perpendicular to the surface of a sphere of radius R. The initial attempt involved taking the cross product with the normal vector, but it was deemed too difficult in spherical coordinates. An alternative method is sought, potentially involving the field being a multiple of the radius vector at each point on the sphere.
  • #1
I'm trying to figure out if a given vector field is perpendicular at the surface of a sphere of radius R. The vector field is given in spherical coordinates.

I initially attempted to take the cross product of the vector field with the normal vector at the surface of the sphere to see if it was zero, but unfortunately, the cross product in spherical coordinates is much too difficult to work with. I'm hoping to find an alternative method.

Does anyone know how to find out if a vector field is perpendicular to a surface without using the cross product?
 
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  • #2
At each point on the sphere the field would have to be a multiple of the radius vector. Something like

[tex]\vec F = f(\theta,\phi)\vec \rho[/tex]

where f is a scalar.
 
  • #3
Thank you very much!
 

1. What is a vector field?

A vector field is a mathematical function that assigns a vector to each point in a given space. It is used to represent and study physical phenomena that involve the concept of a vector, such as velocity, force, and electric and magnetic fields.

2. Why is it important to find a vector field perpendicular to the surface of a sphere?

It is important to find a vector field perpendicular to the surface of a sphere because it can help us understand the behavior of physical quantities, such as force or electric fields, on the surface of a sphere. This information is crucial in many fields of science, including physics, engineering, and astronomy.

3. How do you find a vector field perpendicular to the surface of a sphere?

To find a vector field perpendicular to the surface of a sphere, you can use the concept of gradient. The gradient of a scalar function at a point is a vector that points in the direction of the steepest increase of the function at that point. By taking the gradient of a scalar function that represents the surface of a sphere, we can find a vector field that is perpendicular to the surface at every point.

4. Can a vector field be perpendicular to the surface of a sphere at every point?

No, a vector field cannot be perpendicular to the surface of a sphere at every point. This is because a sphere is a curved surface and the direction of the normal vector, which is perpendicular to the surface, changes at every point. However, it is possible to find a vector field that is perpendicular to the surface of a sphere at a specific point or along a specific curve on the surface.

5. What are some real-life applications of finding a vector field perpendicular to the surface of a sphere?

Finding a vector field perpendicular to the surface of a sphere has many real-life applications. For example, it is used in navigation systems to calculate the direction of travel on the surface of a globe. It is also used in designing structures, such as domes, which require support in all directions. In physics, it is used to study the behavior of electric and magnetic fields on the surface of a sphere, which has applications in electronics and telecommunications.

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