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

Explanation

X and Y are two long straight parallel conductors carrying currents `I_(1) " and "I_(2)` respectively, and placed close to each other. d is the separation between the two conductors and L is the length of the conductors.
The magnetic field at any point on the conductor Y due to current `I_(1)` in the conductor X, is given by `B_(1)=mu_(0)/(4pi)(2I_(1))/d*vec(B)_(1)` acts in a direction perpendicular to the plane containing the two conductors. The conductor Y which carries current `I_(2)`, experiences a mechanical force due to `B_(1)` acting on it and this force is given by `F_(1)=B_(1)L_(2)Lsintheta`.
`therefore " "F_(1)=(mu_(0)/(4pi)(2I_(1))/d)I_(2)Lsintheta`
`" "F_(1)=mu_(0)/(4pi)(2I_(1)I_(2))/dL " ...(1)"`
According to Fleming's left hand rule, the direction of the force `F_(1)` on X is perpendicular to `B_(2)` and is towards the conductor Y.
The magnetic field at any point on the conductor X due to the current `I_(2)` in y, is given by `B_(2)=mu_(0)/(4pi)(2I_(2))/d.vec(B)_(2)` acts on X and opposite to `B_(1)`. The mechanical force acts on X due to `B_(2)` is `I_(1)Lsintheta`.
`therefore " "F_(2)=(mu_(0)/(4pi)(2I_(2))/d)I_(2)Lsin90^(@)`
`" "F_(1)=mu_(0)/(4pi)(2I_(1)I_(2))/dL " ...(2)"`
According to Fleming's left hand side rule, the direction of the force `F_(2)` on X is perpendicular to X and it is towards the conductor y if the current `I_(1)` is inwards (or away from the conductor Y if the current `I_(1)` is outwards)
The Force `F_(1)` acting on a certain length of the conductor Y due to the current in the conductor X is equal in magnitude to the force `F_(2)` acting on the same length of X due to the current in conductor Y. If the two conductors carry the currents in the same direction (parallel currents) then the forces attract each other. If the two conductors carry the currents in the opposite directions (anti parallel currents), then they are found to repel each other because the two forces act away from each other.
The force per unit length on each conductor is
`" "F_(L)=F_(1)/L=F_(2)/L=(mu_(0)/(4pi)(2I_(1)I_(2))/dL)/L`
i.e., `" "F_(L)=mu_(0)/(4pi)(2I_(1)I_(2))/d`
One ampere of current can be defined as that constant current which when maintained through each of the two infinitely long straight parallel conductors of negligible area of cross section in the same direction placed 1 metre apart in vaccum, causes an attractive force of `2 times 10^(-7)Nm^(-1)` length on each conductor.