Coulomb’s law: "The magnitude of the electric force, F, that a charged particle, q, exerts on another particle, Q, is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles." Mathematically:
,
where is the permittivity of free space, ≈ 9×10-12 C²/(N.m²). Coulomb's law applies to elementary particles and small charged objects as long as their sizes are much less than the distance between them. It is also applies to uniform spherical shells or spheres of charge. In that case, r, the distance between the centers of the spheres, must be larger than the sum of the sphere radii; that is to say, the charge must be largely separated.
The electrostatic force between two point charges can be expressed in vector form. The force on due to is written as:
where is a unit vector directed from to and r12 is the distance between them..
When more than one point charge is present, the total electrostatic force exerted on the ith charge is the vector sum of the forces exerted on that charge by others individually, i.e.
where is a vector directed from the charge, , to the point in question.
Electric Field: To calculate the electric field E, it is often convenient to make use of a fictitious charge called a test charge. This charge is similar to a real charge except in one respect: The test charge is defined to exert no force on other charges. It therefore does not disturb the charges in the vicinity. In practical situations, a test charge can be approximated by a charge of nearly negligible magnitude. The test charge will feel an electric force F. The electric field at the location of the point charge is defined as the force F divided by the charge :
The definition of the electric field shows that the electric field is a vector field: the electric field at each point has a magnitude and a direction. The direction of the electric field is the direction in which a positive charge placed at that position will move.
The electric flux (to flow) (Φ, [Φ] = N.m²/C) is represented by the number of electric field lines that penetrate a surface. If the electric field, E, is uniform and makes an angle θ with the normal to the surface area A, the electric flux through the surface is
In general case
Ex.- A uniform electric field E = a i + b j intersects a surface of area A. Calculate the flux through this area if the surface lies:
a)in the XZ-plane. [Answer: ΦXZ= (a i + b j).(A j) = bA.]
b)in the YZ-plane. [Answer: ΦYZ= (a i + b j).(A i) = aA.]
c)in the XY-plane. [Answer: ΦXY= (a i + b j).(A j) = 0.]
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Gauss' law is a very powerful theorem which relates any charge distribution, having a high degree of symmetry, to the resulting electric field at any point in the vicinity of the charge. It states that "the net electric flux, Φc , through any closed surface* is equal to the net charge inside the surface, qin, divided by ɛo". In symbols
,
*usually called Gaussian surface, which has the exact symmetry as the charge distribution.
If the charge has symmetry, such as spherical, cylindrical, etc. one can use
Example 1: Field of point charge. Calculate the electric field due to a point chargeq.
The field generated by a point charge q is spherical symmetric, and its magnitude will depend only on the distance r from the point charge. The direction of the field is along the direction (see Figure). Consider a spherical surface centered around the point charge q (see Figure). The direction of the electric field at any point on its surface is perpendicular to the surface and its magnitude is constant. Using Gauss's law we obtain the following expression
which is Coulomb's law.
Ex. 2- Calculate the electric field due to an insulating sphere of radius R. The sphere has a uniform charge density ρ and total charge Q, where
Ex. 3- Calculate the electric field due to a non-conducting plane sheet of charge Q and has a surface area , .
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