Question

What is the source of force acting on a current-carrying conductor placed in a magnetic field ? Obtain the expression for force acting between two long straight parallel conductors carrying steady currents and hence define 'ampere'.

Hint:

Force per unit length is \(\mu_0 I_1 I_2/(2\pi d)\). For 1 A in each wire at 1 m the force is \(2\times10^{-7}\) N per metre.

Answer 1

Step 1: Understanding the Concept:
A conductor carries current because its free electrons drift. Each moving charge \(q\) in a magnetic field feels the Lorentz force \(q\,\vec{v}\times\vec{B}\). These forces are passed to the lattice of the conductor, so the whole wire feels a force. So the source of the force on a current-carrying conductor is the magnetic force on the moving (drifting) charge carriers in it. For a straight wire of length \(L\) this gives \(\vec{F} = I\,\vec{L}\times\vec{B}\).

Step 2: Field of one conductor at the other:
Let two long straight parallel conductors \(a\) and \(b\) be a distance \(d\) apart in vacuum. They carry currents \(I_a\) and \(I_b\) in the same direction.

The field of \(a\) at the position of \(b\) has magnitude
\[ B_a = \frac{\mu_0 I_a}{2\pi d} \]
Its direction is perpendicular to the plane containing the two wires (by the right-hand rule).

Step 3: Force on conductor b:
A length \(L\) of conductor \(b\) lies at right angles to \(\vec{B}_a\). So the force on it is
\[ F_{ba} = I_b\,L\,B_a = \frac{\mu_0 I_a I_b}{2\pi d}\,L \]
The force per unit length is
\[ f = \frac{F}{L} = \frac{\mu_0 I_a I_b}{2\pi d} \]
Using \(\vec{F} = I\,\vec{L}\times\vec{B}\), the force on \(b\) points towards \(a\). By the same argument the force on \(a\) points towards \(b\), and it has the same size, as Newton's third law needs. So parallel currents attract. Antiparallel currents repel.

Step 4: Definition of the ampere:
Put \(I_a = I_b = 1\ \text{A}\) and \(d = 1\ \text{m}\):
\[ f = \frac{4\pi \times 10^{-7} \times 1 \times 1}{2\pi \times 1} = 2 \times 10^{-7}\ \text{N m}^{-1} \]
So one ampere is that steady current which, when flowing in each of two infinitely long straight parallel conductors of negligible cross-section placed 1 m apart in vacuum, produces a force of \(2\times10^{-7}\) N per metre of length between them.

Final Answer:
The source of the force is the magnetic force on moving charges. The force per unit length is as below, and 1 A is the current that gives \(2\times10^{-7}\) N/m at 1 m separation. \[ \boxed{\dfrac{F}{L} = \dfrac{\mu_0 I_a I_b}{2\pi d}} \]