Two current carrying long wires are placed parallel to each other and separated by a distance d. If the force on the segment of the wires is F, then the force F is related to d as:

CONCEPT:

The force between two parallel currents:

• We know that there exists a magnetic field due to a conductor carrying a current.
• And an external magnetic field exerts a force on a current-carrying conductor.
• Therefore we can say that when two current-carrying conductors placed nearby each other will exert (magnetic) forces on each other.
• In the period 1820-25, Ampere studied the nature of this magnetic force and its dependence on the magnitude of the current, on the shape and size of the conductors, as well as, the distances between the conductors.
• Let two long parallel conductors a and b separated by a distance d and carrying (parallel) currents Ia and Ib respectively.
• The magnitude of the magnetic field intensity due to wire a, on the wire b is,

\(⇒ B_a=\frac{\mu_oI_a}{2\pi d}\)

• The magnitude of the magnetic field intensity due to wire b, on the wire a is,

\(⇒ B_b=\frac{\mu_oI_b}{2\pi d}\)

• The conductors 'a' and ‘b’ carrying a current Ia and Ib respectively will experience sideway forces due to magnetic field Bb and Ba respectively

EXPLANATION:

• If two current-carrying long wires A and B are placed parallel to each other and separated by a distance d. Then the force F on the segment L of the wires is given as,

\(\Rightarrow F=\frac{\mu_o I_aI_b}{2\pi d}L\)     -----(1)

Where Ia = current in the wire A, Ib = current in the wire B, and d = distance between the wires

If I_a and I_b are constant, then,

\(\Rightarrow F\propto\frac{1}{d}\)

• Hence, option 3 is correct.

Additional Information

• The magnitude of the force on the segment L of the wire 'a' due to magnetic field Bb is given as,

\(⇒ F_{ab}=\frac{\mu_o I_aI_b}{2\pi d}L\)

• The magnitude of the force on the segment L of the wire 'b' due to magnetic field Ba is given as,

\(⇒ F_{ba}=\frac{\mu_o I_aI_b}{2\pi d}L\)

• The magnitude of the force per unit length on wire a and wire b is given as,

\(⇒ f_{ab}=f_{ba}=\frac{\mu_o I_aI_b}{2\pi d}\)

Important Points

• When the current flows in the same direction in the two parallel wires then both wires attract each other and if the current flows in the opposite direction in the two parallel wires then both wires repel each other.