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No charge is enclosed by the sphere. Electric flux through its surface is zero. Gauss's law may be expressed as: [6] = where Φ E is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within V, and ε 0 is the electric constant.
The flux through each patch is equal to the normal (perpendicular) component of the field, the dot product of F(x) with the unit normal vector n(x) (blue arrows) at the point x multiplied by the area dS. The sum of F · n, dS for each patch on the surface is the flux through the surface. Here are 3 definitions in increasing order of complexity.
The net electric flux Φ E is the surface integral of the electric field E passing through Σ: =, The net electric current I is the surface integral of the electric current density J passing through Σ : I = ∬ Σ J ⋅ d S , {\displaystyle I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,} where d S denotes the differential vector ...
The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the divergence of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by S equals the volume rate of flux through S. This is the divergence theorem. [2]
This means by definition that the connection ∇ is flat there. In mentioned Aharonov–Bohm effect, however, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically ...
A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. [1]
For simplicity in calculations it is often convenient to consider a surface perpendicular to the flux lines. If the electric field is uniform, the electric flux passing through a surface of vector area A is = = , where E is the electric field (having the unit V/m), E is its magnitude, A is the area of the surface, and θ is the angle between ...
Circulation can be related to curl of a vector field V and, more specifically, to vorticity if the field is a fluid velocity field, =.. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, [4] = = =