Homework Helper gracy said:
The concepts expressed in mathematical terms often imply considerable mathematical sophistication to work the problems. This is almost always not the case. In general; if you are involved in excessive mathematical manipulations you are doing the problem wrong or do not see an easy way to apply Gauss' Gauss law.
Remember that Gauss' law represents the density of flux lines through an area. Remember also that most of the difficult looking integrals never occur. Start with a definition of flux as This states that the flux is the sum of the vector dot product of electric field over a surface times the area.
E represents the electric field, and ds is a vector normal to the surface representing a differential element of surface. The dot product is the component of E parallel to ds, that is, normal to the surface, times ds.
If E is a constant most often the case then the sum of integral of E. You may at this point want to review the definition of dot product in Chapter 1, Gauss law, and the concept of the integral in the Introduction, Mathematical Background. The surface area ds is represented by a vector normal to the surface.
The vector product E. Integrating this product This represents the flux through the surface. This integral of a vector product that sounds like a very difficult problem is performed rather easily with an understanding of dot products and integrals.
Electric field lines are a convenient concept or construct to help us visualize the electric field. The electric field lines are envisioned as vectors going from positive charges to negative charges, the direction a unit positive charge would move.
The electric field lines between two oppositely charged plates are just lines going from one plate to another. Higher fields would be envisioned as more lines or more lines per cross sectional area.
In the case of a sphere surrounding a unit positive charge the field lines would be pointing radially out, the direction a unit positive charge would move.
On spheres with different radii the electric field would be different; the field line density number per square meter would be different.
However if we could sum all the electric field lines over the different spheres we would find the same number of lines. Their number density would decrease as the radii were increased but the total number over a sphere would remain the same. The formal Gauss' law connects flux to the charge contained again via an integral The charge q is the net charge enclosed by the integral.
Place a charge q at the center of a sphere and apply Gauss' law Fig. The sphere we are integrating over in this instance, not a physical sphere, is known as a Gaussian surface a surface where the integral is taken when applying Gauss' law.
Now E is a constant over the surface symmetry so the integral becomes and is just the surface area of the sphere or and 26 3 which is just the statement we have from Coulomb's law and the definition of the electric field.
Calculate the sign and magnitude of the charge on the earth. Gauss' law states that E. The integral of E. Again notice that this integral over the Gaussian surface is quite easy.
First calculate the E fields entering and exiting the cube. Watch the algebraic signs carefully. There is more flux leaving the cube than there is entering it, so the net charge inside the cube must be positive. On the left side E is to the right and the vector representing ds is to the left, so Remember that the direction of ds is always outward from the enclosed volume.
On the right side E is to the right and the vector representing ds is to the right so The integrals over y and z are zero and the integrals over x add to 1.
If the field at the edges of a stream of square cross section 1. Assume that the E field in the direction of the stream is constant. The field is directed in, so the charge is negative and reduces to calculating four identical integrals.
The charge in a cube 1. Find the sign and density of the charge density in this region.Sep 19, · Use Gauss' law to find the electric field once again. gracy said: ↑ 2)what is the difference between the positions p3 and p4 because my textbook states at P3 electric field would have some magnitude but at P4 electric field would be zero because it lies inside the conductor then why P3 is not considered to be inside the conductor?
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. Gauss's law (or Gauss's flux theorem) is a law of physics.
The law is about the relationship between electric charge and the resulting electric field. In words, Gauss's law states that: The net electric flux through any closed surface is equal to 1. Media in category "Gauss' Law" The following 34 files are in this category, out of 34 total.
Gauss's Law The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.
The electric flux through an area is defined as the electric field multiplied by the area of the surface projected in . The left hand side of Gauss’ law becomes E times the surface of the shape you chose.
A Spherical surface becomes, where is the radius of the sphere. A Cylindrical surface becomes, where and are the length and radius of the cylinder.