Question

Deduce the formula of force acting between two parallel current carrying conductors.

Hint:

Find the field of one wire at the other, \(B=\mu_0 I/2\pi d\), then use \(F=BIl\) on the second wire. The force per unit length is \(\mu_0 I_1 I_2/2\pi d\).

Answer 1

Step 1: Look at the force on the first wire instead.
Take the same two long parallel wires separated by \(d\), carrying \(I_1\) and \(I_2\). This time compute the field made by the second wire at the location of the first.

Step 2: Field of the second wire.
Using Ampere's circuital law for a long straight wire, the field produced by current \(I_2\) at distance \(d\) is \(B_2 = \dfrac{\mu_0 I_2}{2\pi d}\), directed perpendicular to the first wire.

Step 3: Force on the first wire.
A length \(l\) of the first wire carrying \(I_1\) sits in this field at right angles, so it experiences \(F_1 = B_2 I_1 l = \dfrac{\mu_0 I_2}{2\pi d}\,I_1\,l = \dfrac{\mu_0 I_1 I_2 l}{2\pi d}\).

Step 4: Force per unit length.
Dividing by the length, \(\dfrac{F_1}{l} = \dfrac{\mu_0 I_1 I_2}{2\pi d}\). This is identical to the force per unit length on the second wire, confirming the mutual (action-reaction) nature of the force.

Step 5: Definition of the ampere.
Putting \(I_1 = I_2 = 1\) A and \(d = 1\) m gives \(\dfrac{F}{l} = \dfrac{\mu_0}{2\pi} = 2\times10^{-7}\) N/m. Thus one ampere is that steady current which, in two infinitely long parallel wires 1 m apart, produces a force of \(2\times10^{-7}\) N per metre of length.

\[\boxed{\dfrac{F}{l} = \dfrac{\mu_0 I_1 I_2}{2\pi d}}\]