IEMDaily - Video Lecture Notes

Formula

Electric field intensity due to a charged conducting sphere

For r>R

E = \frac{Q}{4 π ϵ_{0} r^{2}}

For r<R

E = 0

where r is the distance from he centre of a sphere of radius R

Diagram

Plot of Electric field intensity due to a charged conducting sphere

Diagram

Potential due to a charged conducting sphere

Definition

Solid angle

A solid angle (symbol:

Ω

) is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point.
An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angle's vertex, that the object covers.

Ω = \frac{S}{r^{2}}

Definition

Electric flux

Electric flux is the measure of flow of the electric field through a given area. Electric flux is proportional to the number of electric field lines going through a normally perpendicular surface.

Definition

Define and calculate electric flux density and identify its unit

The Electric Flux Density (D) is related to the Electric Field (E) by:

D = ϵ E

-----[Equation 1]
In Equation [1],

ϵ

is the permittivity of the medium (material) where we are measuring the fields.
The Electric Field is equal to the force per unit charge:

E = \frac{q_{1}}{4 π ϵ R^{2}}

------[Equation 2]
Then the Electric Flux Density is:

D = ϵ E = \frac{q_{1}}{4 π R^{2}}

--------[Equation 3]
From Equation [3], the Electric Flux Density is very similar to the Electric Field, but does not depend on the material in which we are measuring (that is, it does not depend on the permittivity. Note that the

D

field is a vector field, which means that at every point in space it has a magnitude and direction.
The Electric Flux Density has units of Coulombs per meter squared [

C / m^{2}

Definition

Dependence of flux on the inclination

θ

between E and n

The inclination

θ

between

\vec{E}

and

\hat{n}

is a measure of the number of field lines crossing the area element in which the electric field acts. Thus, the number of field lines crossing

A

is proportional to

E A \cos θ

Definition

Relation between electric field lines and flux across a surface

The field lines carry information about the direction of electric field at different points in space. Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of the electric field at those points. The field lines are crowded where the field is strong and are spaced apart where it is weak.
While electric flux is the measure of the flow of the electric field through a given area. It is proportional to the number of electric field lines passing normally through a perpendicular surface.

Definition

Flux of a vector field

Electric flux

ϕ

through an area element

d \vec{S}

is defined by

ϕ = \vec{E} . d \vec{S} = E . d S c o s θ

, which is proportional to the number of field lines cutting the area element. The angle

θ

here is the angle between

\vec{E}

and

d \vec{S}

Result

Electric flux through a closed surface in uniform electric field

According to Gauss's Law, flux through a closed surface is given by:

ϕ = \int \vec{E} . \vec{d} S = \frac{q_{e n c l o s e d}}{ε_{o}}

Since electric field is uniform, it is created by a source very far from the closed surface. Or there is no charge enclosed within the closed surface. Hence, net flux through it is zero.
Note:
This argument does not hold true for an open surface as an open surface can be arbitrarily extended to a closed surface enclosing a non-zero charge in which case the electric flux through the surface may become non-zero.