Name: Magnetic Force Between Two Parallel Wires
Uploaded: 2019-07-19
Duration: 4 min 48 s
Description: A current-carrying wire kept in magnetic field experiences a force. If we keep two current-carrying wires parallel to each other, will they experience any fo...

Question

Derive an expression for the force between two parallel straight conductors carrying currents in the same direction. Hence define one ampere.

Accepted Answer

Consider two parallel long wires $1 and 2$ placed at a distance $d$ and carrying currents $I_{1} and I_{2}$ in the same direction.The conductor $1$ induces a magnetic field $B_{1}$ , which is perpendicular to the plane of the paper and directed vertically downwards which is given by,

$B_{1} = \frac{μ_{0}}{4 π} \frac{2 I_{1}}{d}$ .

The unit length of a conductor $2$ carrying current $I_{2}$ will experience a force $F_{2}$ due to field $B_{1}$ , produced by current $I_{2}$ is given by,

$F_{2} = I_{2} \times 1 \times B_{1} = I_{2} B_{1}$ .

Substituting the value of $B_{1}$ we get $F_{2} = \frac{μ_{0}}{4 π} \frac{2 I_{1} I_{2}}{d}$ . The direction is given by Flemings' left-hand rule. Similarly, the force experienced by the conductor $1$ By the field $F_{2}$ is directed towards

The conductor ( $2$ ). Therefore, they attract each other. We can say,

$F_{1} = - F_{2}$

The force experienced by two parallel long conductors per meter of its length when both carrying currents $I_{1} and i_{2}$ and distant $1 m$ apart is given by,

$F = \frac{μ_{0}}{4 π} (\frac{2 I_{1} I_{2}}{1})$ , Substituting $I_{1} = I_{2} = 1 A$ , we get,

$F = \frac{μ_{0}}{4 π} \times \frac{2 \times 1 \times 1}{1} = 2 \times 10^{- 7} {Nm}^{- 1}$

Hence, Ampere is that constant current which, when maintained in two straight parallel conductors of negligible circular cross-section, of infinite length and placed one meter apart in vacuum, would produce between these two conductors a force equal to $2 \times 10^{- 7} N$ per meter of length.

Derive an expression for the force between two parallel straight conductors carrying currents in the same direction. Hence define one ampere.

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