How do you find the flux of a sphere?
We now find the net flux by integrating this flux over the surface of the sphere: Φ=14πϵ0qR2∮SdA=14πϵ0qR2(4πR2)=qϵ0. Φ=qϵ0. A remarkable fact about this equation is that the flux is independent of the size of the spherical surface.
Is divergence the same as flux?
Divergence and flux are closely related – if a volume encloses a positive divergence (a source of flux), it will have positive flux. “Diverge” means to move away from, which may help you remember that divergence is the rate of flux expansion (positive div) or contraction (negative div).
Can you use divergence theorem on a sphere?
We cannot apply the divergence theorem to a sphere of radius a around the origin because our vector field is NOT continuous at the origin.
What is flux in sphere?
Considering a Gaussian surface in the form of a sphere at radius r > R , the electric field has the same magnitude at every point of the surface and is directed outward. The electric flux is then just the electric field times the area of the spherical surface.
Can you use divergence theorem on an open surface?
Surface must be closed But unlike, say, Stokes’ theorem, the divergence theorem only applies to closed surfaces, meaning surfaces without a boundary. For example, a hemisphere is not a closed surface, it has a circle as its boundary, so you cannot apply the divergence theorem.
Why can’t we use the divergence theorem to evaluate flux integral?
Because this is not a closed surface, we can’t use the divergence theorem to evaluate the flux integral. However, if we had a closed surface, for example the second figure to the right (which includes a bottom surface, the yellow section of a plane) we could. We’ll consider this in the following. The divergence theorem says
What is the divergence at p in this equation?
This equation says that the divergence at P is the net rate of outward flux of the fluid per unit volume. The divergence theorem translates between the flux integral of closed surface S and a triple integral over the solid enclosed by S.
How do you find the divergence of an integral?
The divergence theorem part of the integral: Here div F = y + z + x. Note that here we’re evaluating the divergence over the entire enclosed volume, so we can’t evaluate it on the surface.
What is the use of divergence theorem in physics?
The divergence theorem has many uses in physics; in particular, the divergence theorem is used in the field of partial differential equations to derive equations modeling heat flow and conservation of mass. We use the theorem to calculate flux integrals and apply it to electrostatic fields.