Two infinitely long straight parallel conductors carrying currents I1 and I2 are held at a distance d apart in vacuum. The force F on a length L of one of the conductors due to the other is _______.

The correct option is: proportional to I₁ × I₂ × L

Step-by-Step Explanation:

Consider two long parallel conductors carrying currents I₁ and I₂ separated by a distance d in vacuum.

First, let us calculate the magnetic field B₁ produced by the first conductor (carrying current I₁) at the position of the second conductor (at a distance d):

$$B_1=\frac{{\mu }_0{I}_1}{2\pi d}$$

where μ₀ is the permeability of free space.

The magnetic force F experienced by a length L of the second conductor carrying current I₂ placed in this magnetic field B₁ is given by the formula:

$$F=I_{2}L{B}_1$$

Substituting the value of B₁ into the force equation:

$$F=I_{2}L\left(\frac{{\mu }_0{I}_1}{\mathrm{2<!--\; <mi-->\pi \; <mi>d</mi>}}\right)$$

$$F=\frac{{\mu }_0{I}_1{I}_{2}L}{\mathrm{2<!--\; <mi-->\pi \; <mi>d</mi>}}$$

From this derived formula, we can see that the force F is directly proportional to the product of the currents (I₁ × I₂) and the length L of the conductor, while being inversely proportional to the distance d between them.

Therefore, we can conclude that:

$$F\propto I_1\times I_2\times L$$