Question

Derive the formula for the force acting between two parallel current carrying conductors.

Answer 1

Hint: Before we proceed to derive the expression for force acting between two current carrying parallel conductors, we should be familiar with the expression for the Biot-Savart’s law which gives the expression for magnetic field at a point due to a current carrying conductor, as follows –
$dB={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{I\cdot dl}{r^3}}$
where I = current in the circuit, $dl$= length of the current element, r = distance of the point from the current element.

Complete step by step solution:
Consider two conductors A and B of infinite length, with currents $I_A$ and $I_B$ in the same direction and separated at a distance of d, as shown:

The current $I_A$ results in a magnetic field $B_A$ at the conductor B, at the distance of d, which is given by applying the Biot-Savart’s law for a conductor carrying current.
$B_A={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{2I_A}{d}}$
Whenever a current carrying conductor is placed in an external magnetic field, there is a force acting on it and the force experienced by the conductor is equal to the product of magnetic field, current and the length of the conductor.
$F=BIL$
The conductor B experiences force due to the magnetic field produced by the conductor A, which is given by $B_A$.
Therefore, the force acting on the conductor B will be equal to –
$\Rightarrow F=B_A{I}_{B}L$
Substituting the value $B_A$ , we have –
$\Rightarrow F={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{2I_A{I}_B}{d}}L$
Therefore, the force per unit length between two parallel conductors is given by the expression:
$\Rightarrow f={\displaystyle \frac{F}{L}}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{2I_A{I}_B}{d}}$
The direction of the force can be calculated by using the Fleming’s Left Hand rule where:
i) Index finger represents the direction of the magnetic field, B
ii) Middle finger represents the direction of the current, I
iii) Thumb represents the direction of the force, F
The direction of the force is represented in the diagram below:

Here, the direction of $B_A$is calculated by the Right-Hand screw rule, which gives the result that the direction of the magnetic field is inside the plane of paper and denoted by X.

Note: The standard definition of the SI unit of current, ampere is defined by the help of the derived expression as follows:
If $I_A=I_B=1A$ and $d=1m$, we have –
$\Rightarrow f={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{2I_A{I}_B}{d}}={10}^{-7}\times {\displaystyle \frac{2\times 1\times 1}{1}}=2\times {10}^{-7}N$
Thus, One ampere is defined as the steady current, which when maintained between two infinitely long conductors with unit separation, would produce a force per unit length equal to $2\times {10}^{-7}N$.