Obtain the formula for force acting between two parallel straight… - Vidyadip

Question

Obtain the formula for force acting between two parallel straight current carrying conductors and hence define one ampere.

✓

Answer

→ As shown in the figure, two long parallel wires $a$ and $b$ separated by a distance $d$ and carrying currents $I _a$ and $I _b$ respectively.

→ A uniform magnetic field $\overrightarrow{ B }_a$ is produced by the conductor ' $a$ ' at every point on the conductor ' $b$ '
→ The right-hand rule indicates that the direction of the field is downwards.
→ From Ampere's circuital law,
$
B_a=\frac{\mu_0 I_a}{2 \pi d}
...(1)$
→ This magnetic field causes a force on wire $b$.
→ To calculate this force, a segment is considered on the wire $b$ as shown in the figure. The length of the segment is $L$.
→ The force acting on this segment is $F _{b a}= I _b LB _a$
Putting the value from equation (1)
$
\begin{array}{l}
F_{b a}=I_b L\left(\frac{\mu_0 I_a}{2 \pi d}\right) \\
\therefore F_{b a}=\frac{\mu_0 I_a I_b L}{2 \pi d}
...(2)\end{array}
$
→ The direction of this force is from $b$ to $a$
→ Similarly, the force exerted by wire $b$ on wire $a$ of a segment of length $L$ is
$
F_{a b}=\frac{\mu_0 I_a I_b L}{2 \pi d}
...(3)$
→ The direction of this force is from $a$ to $b$.
→ Thus, $\overrightarrow{ F }_{a b}=-\overrightarrow{ F }_{b a}$
→ Both the forces are equal in magnitude and opposite in direction. This force is attractive.
→ So, if current flows in the same direction in both wires, the force is attractive and if current flows in opposite direction, the force is repulsive.

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