Question
Derive the expression for force per unit length between two long straight parallel current carrying conductors. Hence define one ampere.
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Answer
Figure shows two long parallel conductors a and b separated by a distance d and carrying (parallel) currents $I_a$ and $I _b,$ respectively. The conductor ‘a’ produces, the same magnetic field $B_a$ at all points along the conductor ‘b’. The right-hand rule tells us that the direction of this field is downwards (when the conductors are placed horizontally). Its magnitude is given by
$\text{B}_{\alpha} = \frac{\mu_\circ\text{I}_{\alpha}}{2\pi\text{d}}$
The conductor ‘b’ carrying a current $I _b$ will experience a sideways force due to the field $B_a$. The direction of this force is towards the conductor ‘a’. We label this force as $F_{ba}$, the force on a segment L of ‘b’ due to ‘a’. The magnitude of this force is given by
$\int_{ba} = \text{I}_{b}\text{LB}_{a}$ $ = \frac{\mu_\circ\text{I}_{a}\text{I}_{b}}{2\pi\text{d}}\text{L}$
Let $\int_{ba}$ represent the magnitude of the force $\text{F}_{ba}$ per unit length.
$\int_{ba} = \frac{\mu_\circ\text{I}_{a}\text{I}_{b}}{2\pi\text{d}}$
One ampere: The ampere is the value of that steady current which, when maintained in each of the two very long, straight, parallel conductors of negligible cross-section, and placed one metre apart in vacuum, would produce on each of these conductors a force equal to $2 × 10^{-7}$ newton per metre of their length.
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