GAUSS' LAW
Carl Friedrich Gauss
(1777- 1785)
Gauss' law is an alternative to Coulomb's law for expressing the relationships between electric charge and electric field. It was formulated by Carl Friedrich Gauss (1777- 1785), one of the greatest mathematician of all time. Many areas of mathematics bear the mark of his influence, and he made equally significant contributions to theoretical physics.
Gauss' law states that the total electric flux through any closed surface (a surface enclosing a definite volume) is proportional to the total (net) electric charge inside the surface.
We'll start with the field of a single positive point charge q. The field lines radiate out equally in all directions. We place this charge at the center of an imaginary spherical surface with radius R. The magnitude E of the electric field in every point on the surface is given by
At each point on the surface, the resultant E is perpendicular to the surface, and its magnitude is the same at every point. The total electric flux is just the product of the field magnitude E and the total area of the sphere is equal to
The flux is dependent of the radius R of the sphere. It depends only on the charge q enclosed by the sphere.
We can also interpret this result in terms of field lines. Electric field lines is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction
Carl Friedrich Gauss
(1777- 1785)
Gauss' law states that the total electric flux through any closed surface (a surface enclosing a definite volume) is proportional to the total (net) electric charge inside the surface.
We'll start with the field of a single positive point charge q. The field lines radiate out equally in all directions. We place this charge at the center of an imaginary spherical surface with radius R. The magnitude E of the electric field in every point on the surface is given by
At each point on the surface, the resultant E is perpendicular to the surface, and its magnitude is the same at every point. The total electric flux is just the product of the field magnitude E and the total area of the sphere is equal to
The flux is dependent of the radius R of the sphere. It depends only on the charge q enclosed by the sphere.
We can also interpret this result in terms of field lines. Electric field lines is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction
of the electric field vector at that point. Every field line that passes through the smaller sphere also passes through the larger sphere, so the total flux through each sphere is the same.
What is true of the entire sphere is also true of any portion of its surface. In the figure above, an area dA is outlined on a sphere at radius R and then projected the sphere of radius 2R by drawing lines from the center through points on the boundary of dA. The area projected on the larger sphere is clearly 4 dA. But since the electric field due to a point charge is inversely proportional to r², the field magnitude is 1/4 as great on the sphere of radius 2R as on the sphere of radius R. Hence, the electric flux is the same for both areas and is independent of the radius of the sphere.
This projection technique shows us how to extend this discussion to nonspherical surfaces. Instead of the second sphere, let us surround the sphere of radius R by a surface of irregular shape, as in figure 2. Consider a small element of area dA on the irregular surface; we note that this area is larger than the corresponding element on a spherical surface at the same distance from q. If a normal to dA makes an angle with a radial line from q, two sides of the area projected onto the spherical surface are foreshortened by a factor cos ( figure2b). The other two sides are unchanged. Thus, the electric flux E dA cos for each, and sum the results by integrating as in . Each of the area elements projects onto a corresponding spherical surface element. Thus, the electric flux through the irregular surface, given by any forms of equation, must be the same as thge total flux through the sphere, which the equation mentioned earlier showsnis equal to blah blah..
Thus, for irregular surface,
The equation above holds for a surface at any shape or size, provided only that it is surface enclosing the charge closedq. The circle on the integral sign reminds us that the integral is always taken over a closed surface.
The area elements dA and the corresponding unit vectors n always point out of the volume enclosed by the surface. The electric flux is then positive in areas where the electric field points out of the surface and negative where it is inward.
What is true of the entire sphere is also true of any portion of its surface. In the figure above, an area dA is outlined on a sphere at radius R and then projected the sphere of radius 2R by drawing lines from the center through points on the boundary of dA. The area projected on the larger sphere is clearly 4 dA. But since the electric field due to a point charge is inversely proportional to r², the field magnitude is 1/4 as great on the sphere of radius 2R as on the sphere of radius R. Hence, the electric flux is the same for both areas and is independent of the radius of the sphere.
This projection technique shows us how to extend this discussion to nonspherical surfaces. Instead of the second sphere, let us surround the sphere of radius R by a surface of irregular shape, as in figure 2. Consider a small element of area dA on the irregular surface; we note that this area is larger than the corresponding element on a spherical surface at the same distance from q. If a normal to dA makes an angle with a radial line from q, two sides of the area projected onto the spherical surface are foreshortened by a factor cos ( figure2b). The other two sides are unchanged. Thus, the electric flux E dA cos for each, and sum the results by integrating as in . Each of the area elements projects onto a corresponding spherical surface element. Thus, the electric flux through the irregular surface, given by any forms of equation, must be the same as thge total flux through the sphere, which the equation mentioned earlier showsnis equal to blah blah..
Thus, for irregular surface,
The equation above holds for a surface at any shape or size, provided only that it is surface enclosing the charge closedq. The circle on the integral sign reminds us that the integral is always taken over a closed surface.
The area elements dA and the corresponding unit vectors n always point out of the volume enclosed by the surface. The electric flux is then positive in areas where the electric field points out of the surface and negative where it is inward.
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