Derive a mathematical expression for the force per unit length ex… - Vidyadip

Question

Derive a mathematical expression for the force per unit length experienced by each of the two long current carrying conductors placed parallel to each other in air. Hence define one ampere of current.

Explain why two parallel straight conductors carrying current in the opposite direction kept near each other in air repel?

✓

Answer

Diagram:

Magnetic field due to the current $\text{I}_{1}$ flowing in conductor 1 at a point on conductor 2 $\text{B}_{1} = \frac{\mu_\circ\text{I}_{1}}{2\pi\text{d}}$

$\therefore $ force on conductor 2 due to $\text{B}_{1}$ is

$\text{F}_{2} = \text{I}_{2} \big( \overrightarrow\ell{_2}\times\overrightarrow{\text{B}}_{1}\big)$

$=\text{I}_{2}\ell_{2}\times\text{B}_{1}$

$\therefore \frac{\text{F}_{2}}{\ell_2} = \frac{\mu_\circ\text{I}_{1}\text{I}_2}{2{\pi}\text{d}}$

Ampere: The equal currents, flowing through two thin long straight parallel conductors said to be one ampere each if they interact with each other with a force of $2\times 10^{-7}$ N/m when kept one metre apart in vacuum.
Direction of magnetic field, at the second conductor, due to current in the first conductor.

Direction of force on the second conductor, (carrying a parallel current) due to this magnetic field.

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