Obtain an expression for the force between two straight parallel conductor carrying current. Hence define ampere.

Explanation

Consider two infinitely long straight parallel conductors a and b carrying currents `I_(1) and I_(2)` respectively and separated by a perpendicular distance .d. as shown in the figure. The magnetic field at each point on conductor .b. due to current `I_(1)` in conductor .a. is
`B_(1)= (mu_(0))/(4pi) (2I_(1))/(d)`

Now the current carrying conductor .b. is in uniform magnetic field `B_(1)`. Hence magnetic force on the segment L of conductor .b. is
`F_(1)= I_(2) L B_(1) sin theta`
`vec(F)_(1)` is directed towards the conductor .a. and here `theta= 90^(@), sin 90^(@)=1`
`F_(1)= (mu_(0))/(4pi) (2I_(1)I_(2))/(d)L`
Similarly conductor .a. also experience same magnitude of force but in opposite direction. Magnetic force on segment L of conductor .a. is
`F_(2)= (mu_(0))/(4pi) (2I_(1)I_(2))/(d)L`
The mutual force per unit length on conductors .a. and .b. is, `F= (F_(1))/(L) or (F_(2))/(L)`
`therefore F= (mu_(0))/(4pi) (2I_(1)I_(2))/(d)`
Definition of ampere: Thus ..If two very long, straight, parallel conductors of negligible cross section carrying same steady current are placed 1m apart in free space (vacuum) experience a mutual force of `2 xx 10^(-7)` newton per meter length of these conductors, then the current in each conductor is said to be 1A...