Derive an expression for the force per unit length between two infinitely long straight parallel current carrying wires. Hence, define one ampere $( \mathrm{A} )$.
Derive an expression for the force per unit length between two infinitely long straight parallel current carrying wires. Hence, define one ampere $( \mathrm{A} )$.
Two wires are parallel and at $d$ distance from each other carrying current $\mathrm{I}_{a}$ and $\mathrm{I}_{b}$. Magnetic force on L length of $b$ due to $a$ is given by,
$\mathrm{F}_{b a}=\frac{\mu_{0} \mathrm{I}_{a} \mathrm{I}_{b} \mathrm{~L}}{2 \pi d} \quad \ldots \text { (1) }$
Force per unit length $\left(\mathrm{L}=1\right.$ unit) is given by $f_{b a}$ which is represented as,
$f_{b a}=\frac{\mu_{0} I_{a} I_{b}}{2 \pi d} \quad \ldots \text { (2) }$
This formula is applicable to detect SI unit of current and its SI unit is from above equation,
$\mathrm{I}_{a}=\mathrm{I}_{b}=1 \mathrm{~A}$
$d=1 \mathrm{~m}$
$\mu_{0}=4 \pi \times 10^{-7} \frac{\mathrm{T} \cdot \mathrm{m}}{\mathrm{A}}$
Taking all these we get,
$f_{b a}=2 \times 10^{-7} \mathrm{~N}$
Definition of one ampere's : 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 meter apart in vacuum would produced on each of these conductors a force equal to $2 \times 10^{-7} \mathrm{~N}$ per unit meter of length.
An instrument called the current balance is used to measure this mechanical force.
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