Question

The unit of current, the ampere, is defined in terms of the force between currents. Two 1.0-meter-long sections of very long wires a distance 2.0m apart each carry a current of 1.0 A

What is the force between them? (If the force between two actual wires has this value, the current is defined to be exactly 1 A.)

Answer 1

Concepts and reason

The concepts required to solve the given problem is magnetic field due to a current carrying wire and force on a current carrying wire due to magnetic field.

Initially, determine the magnetic field at one wire due to other wire by using expression magnetic field due to a current carrying wire. Then, determine the force on the wire by using magnetic field, length of the wire and current in the wire.

Fundamentals

The current carrying wire creates a magnetic field around the wire and the direction of magnetic field around the wire is known by using right hand rule.

The magnitude of magnetic field B due to a current carrying wire at distance r from the wire is given by following expression.

$B = \frac{{{\mu _0}I}}{{2\pi r}}$

Here, I is the current, and ${\mu _0}$ is the permeability.

The magnetic force F on a current carrying wire is given by following expression.

$F = BIL$

Here, L is the length of the wire.

The magnitude of magnetic field B due to a current carrying wire at distance r from the wire is given by following expression.

$B = \frac{{{\mu _0}I}}{{2\pi r}}$

Substitute 1.0 A for I, 2.0 for r, and $4\pi \times {10^{ - 7}}\;{\rm{H/m}}$ for ${\mu _0}$ in the above equation.

$\begin{array}{c}\\B = \frac{{\left( {4\pi \times {{10}^{ - 7}}\;{\rm{H/m}}} \right)\left( {1.0\;{\rm{A}}} \right)}}{{2\pi \left( {2.0\,{\rm{m}}} \right)}}\\\\ = 1 \times {10^{ - 7}}\,{\rm{T}}\\\end{array}$

The magnetic force F on a current carrying wire is,

$F = BIL$

Substitute 1.0 A for I, $1 \times {10^{ - 7}}\,{\rm{T}}$ for B, and 1.0 m for L in the above equation.

$\begin{array}{c}\\F = \left( {1 \times {{10}^{ - 7}}\,{\rm{T}}} \right)\left( {1.0\,{\rm{A}}} \right)\left( {1.0\,{\rm{m}}} \right)\\\\ = 1 \times {10^{ - 7}}\,{\rm{N}}\\\end{array}$

Ans:

The force between the wires is $1 \times {10^{ - 7}}\,{\rm{N}}{\rm{.}}$