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Show that currents in two long, straight, parallel wires exert forces on each other. Derive the expression for the force.###Derive an expression for the force per unit length between two infinitely long parallel conductors carrying current and hence define the ampere.

Vidyadip · Accepted Answer

When two currents pass in adjacent parallel straight conductors, we may think of each of the currents as being situated in the magnetic field caused by the other current. This results in a force on each conductor.

Consider two infinitely long, straight, parallel wires, each of length ; a distance s apart in vacuum, as shown in figure (a). The magnetic field around the wire 1 , carrying a current $l_1$ has an induction of magnitude
$
B _1=\left(\frac{\mu_0}{4 \pi}\right) \frac{2 I_1}{s}
$
Wire 2 , with a current $I_2$ in the same direction as $l_1$, is situated in this field. The direction of the field
with induction $\overrightarrow{B_1}$ at the position of wire 2, given by the right hand Igripi rule, is
perpendicular to the plane of the two conductors, as shown. Hence, the force $\overrightarrow{F_2}$ on wire 2
has a magnitude
$
F _2= I _2 \mid B _1=\left(\frac{\mu_0}{4 \pi}\right) \frac{2 I_1 I_2 l}{s}
$
and is, by Fleming's left hand rule, towards wire 1. Similarly, the magnetic induction $\overrightarrow{B_2}$ at the position of wire 1 has a magnitude
$
B _2=\left(\frac{\mu_0}{4 \pi}\right) \frac{2 I_2}{s}
$
and is also directed perpendicular to the plane of the wires. Hence, the force on wire 1 has a magnitude
$F _1=I_1 \mid B _2=\left(\frac{\mu_0}{4 \pi}\right) \frac{2 I_1 I_2 l}{s}$
directed towards wire 2 . Thus, the two currents attract each other. $\vec{F}_1=-\vec{F}_2$, i.e, they are equal in magnitude and opposite in direction.
Ampere found that the wires attracted each other when the currents in them were in the same direction [from figure (a)], and repelled each other when they were in the opposite directions [from figure (b)].
From the Eq. (2), the force per unit length acting on each wire is $\frac{F}{l}=\left(\frac{\mu_0}{4 \pi}\right) \frac{2 I_1 I_2}{s}$
Using SI units, $\mu_0 / 4 \pi=10^{-7} N / A ^2$ and, if $I _1= I _2=1 A$ and $s =1 m$, then $\frac{F}{l}=2 \times 10^{-7} N / m$
In $SI$, this equation is the defining relation for the ampere.
Definition: The ampere is that constant current which if maintained in two infinitely long straight parallel wires, and placed one metre apart in vacuum, would cause each conductor to experience a force per unit length of $2 \times 10^{-7}$ newton per metre. [Note : $1 Wb / A \cdot m =1$ $T \cdot m / A =\mid N / A ^2$.]

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