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. ?

To derive the mathematical expression for the force per unit length experienced by two parallel current-carrying conductors, we start by considering the fundamental principles of electromagnetism. When an electric current flows through a conductor, it generates a magnetic field around it. If there are two parallel conductors carrying currents, they will interact with each other's magnetic fields, resulting in a force between them.

Understanding the Magnetic Field Around a Conductor

According to Ampère's circuital law, a long straight conductor carrying a current \( I \) produces a magnetic field \( B \) around it. The magnetic field at a distance \( r \) from a long straight conductor is given by:

B = \frac{\mu_0 I}{2\pi r}

Here, \( \mu_0 \) is the permeability of free space, approximately equal to \( 4\pi \times 10^{-7} \, \text{T m/A} \).

Force Between Two Parallel Conductors

Now, consider two parallel conductors, A and B, separated by a distance \( d \). If conductor A carries a current \( I_1 \) and conductor B carries a current \( I_2 \), the magnetic field produced by conductor A at the location of conductor B is:

B_A = \frac{\mu_0 I_1}{2\pi d}

Conductor B, which is in the magnetic field created by conductor A, will experience a force due to this magnetic field. The force \( F \) on a length \( L \) of conductor B can be calculated using the formula:

F = I_2 L B_A

Substituting the expression for \( B_A \) into this equation gives:

F = I_2 L \left(\frac{\mu_0 I_1}{2\pi d}\right)

Force Per Unit Length

To find the force per unit length \( f \), we divide the total force \( F \) by the length \( L \):

f = \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

This equation shows that the force per unit length between two parallel conductors is directly proportional to the product of the currents flowing through them and inversely proportional to the distance between them.

Defining One Ampere

Now that we have derived the expression for the force per unit length, we can define one ampere. By definition, one ampere is the constant current that, when maintained in two straight parallel conductors of infinite length and placed one meter apart in a vacuum, will produce a force of \( 2 \times 10^{-7} \, \text{N} \) per meter of length between those conductors.

In summary, the relationship between current, force, and distance in this context provides a clear and practical definition of the ampere, linking it to the fundamental forces that arise from electromagnetic interactions.