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’.

Answer 1

Force between two long, straight, parallel conductors carrying current in the same direction:

Two wires AB and CD carry currents I₁ and I₂, respectively, in the same direction.

The distance between them is d.

Due to I₁ current in AB,

Magnetic field at any point on CD is B₁ = `mu_0/(4 pi) xx (2 I_1)/d`

Force on unit length of CD is F₁ = `(BIL)/L` = B₁ × I₂ × 1

= `mu_0/(4 pi) xx (2 I_1 I_2)/d`, acting towards AB    ...[applying Fleming’s left-hand rule]

Similarly,

Due to I₂ current in CD,

Magnetic field at any point on AB is B₂ = `mu_0/(4 pi) xx (2 I_2)/d`

Force on unit length of AB is F₂ = `(BIL)/L` = B₂ × I₁ × 1

= `mu_0/(4 pi) xx (2 I_1 I_2)/d` acting towards CD    ...[applying Fleming’s left-hand rule]

So, the magnitudes of F₁ and F₂ are the same and oppositely directed. So, the forces are attractive. 

Definition of 1 ampere:

If I₁ = I₂ ampere and d = 1 m,

Then, F = `mu_0/(4 pi) xx (2 I_1 I_2)/d`

= 2 × 10⁻⁷ N

So, 1 Ampere may be defined as the amount of steady current which, when maintained in each of the two very long, straight, parallel conductors of negligible cross-section, and placed one metre apart in a vacuum, would exert on each of these conductors a force equal to 2 × 10⁻⁷ N per metre of length.