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: Set up the arrangement.
Consider two long, straight, parallel conductors placed a distance \(d\) apart in air (vacuum). Let them carry steady currents \(I_1\) and \(I_2\) in the same direction.

Step 2: Magnetic field of the first wire at the second.
A long straight wire carrying current \(I_1\) produces, at a perpendicular distance \(d\), a magnetic field
\[ B_1 = \frac{\mu_0 I_1}{2\pi d} \]
By the right-hand rule this field \(B_1\) is perpendicular to the second wire and lies in the plane containing the two wires.

Step 3: Force on a length of the second wire.
The second wire carries current \(I_2\) through this field \(B_1\). The force on a length \(l\) of a current-carrying conductor in a magnetic field is \(F = B I l \sin\theta\). Here the field is perpendicular to the current, so \(\theta = 90^{\circ}\) and \(\sin\theta = 1\):
\[ F_2 = B_1 I_2 l = \frac{\mu_0 I_1}{2\pi d}\, I_2\, l \]
\[ F_2 = \frac{\mu_0 I_1 I_2 l}{2\pi d} \]

Step 4: Force per unit length.
\[ \frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d} \]

Step 5: Direction and meaning.
Using Fleming's left-hand rule, when the two currents flow in the same direction the wires attract each other, and when they flow in opposite directions they repel. By symmetry the first wire feels an equal and opposite force from the field of the second wire, as required by Newton's third law. This relation is the basis of the definition of the ampere.

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