Electromagnetism - Electrical Magnetic Forces

Electrical Magnetic Forces

From coulomb's law the magnitude of electrostatic forces between two charges q₁ and q₂ separated by distance r is given by 

F = 1 / 4π?₀ q₁ q₂ / r²                                                              (1) 

Consider two parallel conducting wires of length dl₁ and dl₂ carrying currents I₁ and I₂ separated by distance.

The magnitude of the magnetic force acting between these two current elements is given by 

F = μ₀ /4π I₁ I₂ / r² dl₁ dl₂                                                       (2) 

let q₁ and q₂ be the charges flowing for time t in wires for current s I₁ and I₂. Then

I₁ dl₁ = q₁ / t d_l1 = d1 v₁ and I₂ dl₂ = d₂ / t dl₂ = q₂ v₂ 

∴ Fm = μ₀ / 4π q₁ q₂ / r₂ v₁ v₂                                            (3)

Dividing by we get 

Fm / Fe = v₁ v₂ (μ₀ ?₀)                                                            (4)

Since left hand side is dimensionless therefore right hand side of must also be dimensionless. It means the quantity μo ?o must have th dimensions of (velocity)-2 here v₁ and v₂ are the drift velocities of electrons incurrent elements As drift velocities are of the order of 10^-5ms ^-1 therefore 

v₁ v₂ = 10^-5 X 10^-5 = 10 ^{- 10} m² s^-2

It is also found that 

C₂ = 1 / μ₀ ?₀ or μ₀ ?₀ = c^-2

Where c is the velocity of light 

C = 3 x 10⁸ ms^-1 
 
Putting values in (17) we get

Fm / Fe = 10^-10 / (3 x 10⁸) 2 = 10^-27

It shows that the magnetic forces are very much smaller than the electrostatic forces in current carrying conductors. However, the electric forces never dominate the magnetic forces in current carrying conductors because 
    
The matter is electrically neutral to very high degree of accuracy. The flowing charges do not disturb the neutrality of the matter. Thus the coulomb electric forces are very rarely in evidence.
    
In an electric current the large numbers of electrons are drifting towards the same direction this adds up weak magnetic fields which becomes evident.

Note if v₁ and v₂ are comparable to the speed of light the magnetic and electric forces are comparable. 

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