Electric Field
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Relative Permittivity
The permittivity of any medium with respect to the permittivity of free space (or vacuum) is called the relative permittivity of that medium. It is denoted by εr and is given by:
The relative permittivity of the medium is also known as the dielectric constant of the medium. It is denoted by K.
Thus, for SI system and in a medium other than air:
OR
Permittivity
Electric Field
Test Charge
The positive charge having unit magnitude is taken as test charge in electrostatics. It is denoted by q0.
Electric Field Intensity
If F is the force experienced by the unit positive test charge q0 at a point in an electric field, the electric field intensity is given by:
E is a vector quantity and its SI unit is N/C.
Electric Field Intensity Due to a Point (Test) Charge
Let us consider a charge +q at point O in space and also consider a point P at distance r from O so that OP = r. If a test charge +q0 is placed at P, the force experienced by the test charge +q0 is given by:
For Vacuum
By definition, the magnitude of electric field intensity E at a point at distance r from charge +q is:
For Other Medium
Electric Flux (φ)
The number of electric lines of force passing through a given surface when field is perpendicular to the direction of line of force is called electric flux.
Mathematically, electric flux is defined as the product of electric field intensity and surface area when the field lines are parallel to the surface area vector.
Where, E = electric field intensity and A = surface area.
In case the surface area vector is perpendicular to the field lines, θ = 90°, then:
There is no flux through the surface parallel to the field.
Gauss’s Theorem
Application of Gauss’s Theorem: Electric Field Due to a Charged Sphere
(i) At a Point Outside the Sphere
Let us consider a point P which is at distance r from the centre of a sphere of radius R at which the electric field intensity is to be determined. For this, draw a Gaussian surface through point P enclosing the charge +q, which is also a sphere of radius r.
If E is the electric field intensity at point P, then electric flux passing through the Gaussian surface is:
Also, from Gauss’s theorem:
From equations (ii) and (iii):
Thus, this is the required expression for electric field intensity due to a charged sphere when the point lies outside the sphere.
(ii) At a Point on the Surface of Sphere
In this case Gaussian surface has radius equal to the charged sphere, i.e. r = R.
From equations (ii) and (iii):
This is the required expression for electric field intensity due to a charged sphere when the point lies on the surface of sphere.
(iii) When Point Lies Inside the Sphere
In this case Gaussian surface does not enclose any charge (i.e. q = 0). If E is the electric field intensity at point P, then the total electric flux passing through the Gaussian surface is given by:
Also, from Gauss’s theorem:
From equations (i) and (ii):
Hence, electric field intensity due to a charged sphere when the point lies inside the hollow sphere must be zero.
Surface Charge Density
The amount of electric charge per unit area of a charged surface of the conductor is called surface charge density. It is denoted by σ.
Electric Field Due to a Charged Plane Conductor
Let us consider a charged plane conductor with uniform surface charge density σ. Let P be the point outside the charged plane conductor about which electric field intensity is to be determined. For this, draw a Gaussian surface of surface area A as shown in the figure.
If E is the electric field intensity, then flux is:
Also, the net charge q enclosed by the Gaussian surface is:
From Gauss’s theorem:
Using equation (ii) in equation (iii):
Comparing equations (i) and (iv):
This is the required expression for electric field intensity due to a charged plane conductor.
Field Outside a Charged Plane Conductor
Let us consider a charged plane conductor with uniform surface charge density σ. Let P be any point outside the charged plane conductor about which electric field intensity is to be determined. For this, draw a Gaussian surface of surface area A as shown in the figure.
From equations (i), (ii) and (iii):
This is the required expression for electric field intensity outside a charged plane conductor.
Linear Charge Density (λ)
Charge per unit length of the conductor is called linear charge density.
Electric Field Intensity Due to Linear Charge Density
Let us consider an infinitely long straight conductor of uniform linear charge density λ. Let P be any point about which electric field intensity is to be determined. For this, draw a Gaussian surface of length l and radius r as shown in the figure.
If E is the electric field intensity, then flux is:
Also, net charge q enclosed by the Gaussian surface is:
From Gauss’s theorem:
From equations (i) and (iii):
This is the required expression for electric field intensity due to linear charge density.
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