Question
Show that currents in two long, straight, parallel wires exert forces on each other. Derive the expression for the force per unit length on each conductor.
✓
Answer
Case I: Both wires carry current in the same direction.
i. Consider two long parallel wires separated by distance $d$ and carrying current $I _1$ and $I _2$ respectively same direction as same as shown in the figure below:
Two long parallel wires, distance $d$ apart
ii. The magnetic field at the second wire due to the current $I_1$ in the first one, according to Biot - Savart's law is $B =\frac{\mu_0 I _1}{2 \pi d } \ldots$..(1)
iii. By the right-hand rule, the direction of this field is into the plane of the paper.
iv. Force on the wire 2 , because of the current $I_2$ and the magnetic field $B$ due to current in wire 1, applying Lorentz force law is,
$
F = I _2\left(\frac{\mu_0 I _1}{2 \pi d }\right) \int dl \ldots(2)
$
The direction of this force is towards wire 1, i.e., it will be an attractive force.
$v$. Force of attraction per unit length of the wire will be $\frac{ F }{ L }=\frac{\mu_0}{2 \pi} \frac{ I _1 I _2}{ d } \ldots$...(3)
Case II: Two wires carry current in opposite direction.
Force is of repulsive nature between antiparallel currents and the magnitude of the force of repulsion per unit length is, $\left|\frac{ F }{ L }\right|=\frac{\mu_0}{2 \pi} \frac{ I _1 I _2}{ d }$
i. Consider two long parallel wires separated by distance $d$ and carrying current $I _1$ and $I _2$ respectively same direction as same as shown in the figure below:
Two long parallel wires, distance $d$ apart
ii. The magnetic field at the second wire due to the current $I_1$ in the first one, according to Biot - Savart's law is $B =\frac{\mu_0 I _1}{2 \pi d } \ldots$..(1)
iii. By the right-hand rule, the direction of this field is into the plane of the paper.
iv. Force on the wire 2 , because of the current $I_2$ and the magnetic field $B$ due to current in wire 1, applying Lorentz force law is,
$
F = I _2\left(\frac{\mu_0 I _1}{2 \pi d }\right) \int dl \ldots(2)
$
The direction of this force is towards wire 1, i.e., it will be an attractive force.
$v$. Force of attraction per unit length of the wire will be $\frac{ F }{ L }=\frac{\mu_0}{2 \pi} \frac{ I _1 I _2}{ d } \ldots$...(3)
Case II: Two wires carry current in opposite direction.
Force is of repulsive nature between antiparallel currents and the magnitude of the force of repulsion per unit length is, $\left|\frac{ F }{ L }\right|=\frac{\mu_0}{2 \pi} \frac{ I _1 I _2}{ d }$
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