Tuesday, October 11, 2011

MATHEMATICAL TREATMENT OF COULOMB'S LAW

Consider two charges of magnitude ๐‘ž₁ and ๐‘ž₂ , assumed to be concentrated as point sources at a distance d from each other.


According to the second law of electrostatics, the force F between them may be expressed mathematically as:

Fq1×qkmfafaie2

F1d2

or        Fq1q2d2

F=kq1q2d2N
                                                      ..................................(1)

where, N is Newton and k is a Constant depending on the nature of the medium between the charges. 

The value of k has been found experimentally, 
k=14ฯ€ฯต0ฯตr
                                                     ....................................(2)
where ,
ฯตr
ฯต0

ฯต0
ฯต0
ฯต0
๐œ–แตฃ is the relative permittivity of the medium intervening the charges
๐œ–₀ is the permittivity of the evacuated free space

The value of ๐œ–₀ in rationalised  M.K.S. unit is 8.854×10⁻¹² farads per meter.

When d is measured in meters, the constant 4๐œ‹ , the proportionality factor for the rationalised M.K.S. units.

Substituting the value of k from Eq. (2) in Eq. (1) we get,

F=q1q24ฯ€ฯต0ฯตrd2N


where,
๐‘ž₁ and ๐‘ž₂ are measured in coulombs
d in meters
๐œ– in farad / meter
F in newtons

For air         ๐œ–แตฃ = 1

∴    F=q1q24ฯ€ฯต0d2N

when     
๐‘ž₁ = ๐‘ž₂ = 1 coulomb
d = 1 meter
๐œ–₀ =  8.854×10⁻¹² farad / meter

F=1×14ฯ€×8.854×1012×12

F=8.997×109



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