PROOF OF GAUSS'S THEOREM

Let a charge of q coulombs be placed inside a hollow enclosure as shown in figure:

Let a small surface area dA of this enclosure subtend a solid angle d𝜔 at q and be at a distance d from the charge q.

The flux density D' (considered in the direction of the radius vector d) will be:

=q4πd2 Coulombs/ sq.meter  or  C/m2

and its component D, normal to the surface will be

=q4πd2cos 𝜃  C/m2

where 𝜃 is the angle between D and D'.

Now, the flux d𝜓 crossing normally the surface of area dA = D ⤬ area



or        dψ=q4πd2cos θ⋅dA    coulombs

By definition,

solid angle=surface arearadius

∴    dω=dA  cos θd2


∴    dψ=q4π⋅dω


or the total flux 𝜓 , crossing the total surface of the imagined enclosure is

∫dψ=q4π ∫dω

ψ=q4π×4π


or      ψ=q coulombs


Thus the total electric flux due to a charge at any point in an enclosure is equal to the charge enclosed in that enclosure.

If there are a number of charges +𝑞₁ , +𝑞₂ , -𝑞₃  and  +𝑞₄ placed in an enclosure, the total electric flux due to all the charges will be

𝜓 = 𝑞₁ + 𝑞₂ - 𝑞₃ + 𝑞₄

or     𝜓 = 𝛴𝑞

PROPERTIES OF LINES OF FORCE

Electrostatic lines of force possess the following properties:

(1) They originate from a positive charge and terminate on a negative charge.

(2) They are always normal to the surface of the body at the point from where they originate or terminate.

(3) A unit positive charge when placed in close neighbourhood of a positively charged body, will follow the path of a line of force and will move to the negatively charged body.

(4) A line of force is such that a tangent to it at any point indicates a direction of the electric intensity at that point.

RELATION BETWEEN B,H, I AND k

When a magnetic material (say iron) of cross-sectional area A, and relative permeability 𝜇ᵣ is placed in an uniform field of intensity H, two types of lines of induction pass through it: one due to the magnetizing field H and other due to the iron piece itself being magnetized by induction.

Thus total flux density B will be given by

B = 𝜇₀ H + I

where       𝜇₀ = permeability of free space

Now absolute permeability 𝜇ₐ is given by



Dividing either side by 𝜇₀ , we get the relative permeability

MAGNETIC SUSCEPTIBILITY (k)

Magnetic Susceptibility (k) : The Ratio between the intensity of magnetization produced in a substance and the magnetizing force producing it, is called the magnetic susceptibility of the substance, and is denoted by k.

i.e.

INTENSITY OF MAGNETIZATION (I or J)

If an iron piece of area of cross-section A sq meter when placed in an uniform magnetic field of intensity H develops a pole strength of (say) m webers, then the intensity of magnetization I (or J) is defined by

As m is the pole strength induced in the iron piece, the total flux of the piece is m and the flux density is also m/A ; in other words, the intensity of magnetization J, or I, of the iron piece may also be defined as the flux density produced in it due to its own induced magnetism

If l is the length of the iron piece, in meters

where M is the magnetic moment of the piece, and V is its volume in cubic meters.

Thus the intensity of magnetization of a substance can also be defined as the magnetic moment developed per unit volume of the substance.

GAUSS'S THEOREM

Gauss Theorem: Gauss theorem states that "The total number of lines of magnetic induction emanating from a pole to any imaginary closed surface is equal to its pole strength."

Let there be a point pole of strength m webers placed inside a closed surface.

By definition, if B be the flux density over the small elementary area dA sq meters, at a distance r meters, from, the point pole then the flux over the elementary area dA is

If 𝜃 is the angle between the normal to the elementary area dA and the direction of flux, then the flux normal to this elementary area dA is given by

The total flux over the enclosed area is

The factor     is the small solid angle dw subtended by the area dA at the pole.

FIELD OF UNIFORM INTENSITY

Field of Uniform Intensity: A magnetic field in which a unit pole experiences the same force at all points but the direction of force varies from point to point is termed as "Field of Uniform Intensity".

UNIFORM MAGNETIC FIELD

Uniform Magnetic Field: An Uniform Magnetic Field is one in which a pole of say unit strength experiences the same force, is the same.

The lines of force are parallel and straight in an uniform field.

A well known example of an uniform field is magnetic field due to the earth.

MAGNETIC FLUX

Magnetic Flux (𝜙):

In the case of a point pole of strength m webers, the flux density B at distance r is

B=m4πr2   webers/sq meter

∴ Total flux 𝜙 emanating out of strength m webers is

ϕ=flux density×surface area  =B×A

 =m4πr2×4πr2=m webers