Question

Two thin long, parallel wires separated by a distance d carry a current i each in the same direction. They will
A. Repel each other with a force per unit length $\dfrac{{{\mu _0}{i^2}}}{{2\pi d}}$
B. Attract each other with a force per unit length $\dfrac{{{\mu _0}{i^2}}}{{2\pi d}}$
C. Repel each other with a force per unit length $\dfrac{{{\mu _0}{i^2}}}{{2\pi {d^2}}}$
D. Attract each other with a force per unit length $\dfrac{{{\mu _0}{i^2}}}{{2\pi {d^2}}}$

Answer 1

Hint: All magnetic fields are due to currents and due to intrinsic magnetic moments of particles. Use the force per unit length between two long current carrying conductor’s formula. Then compare the Magnetic field created by conductor 1 at the location of the conductor 2 and conductor 2 gets attracted towards conductor 1.

Complete step by step answer:
The magnetic field produced due to a current carrying segment is given by Biot Savart law. It is a vector quantity . In order to know the magnetic field produced at a point due to this small element, one can use Biot-Savart’s Law. According to Biot Savart law, the magnetic field depends is directly proportional to the current, is directly proportional to the length and is inversely proportional to the square of the distance of the point from the element.

The force per unit length between two thin long conductors carrying current is given by
$F = \dfrac{{{\mu _0}{i_1}{i_2}}}{{2\pi d}}$
Where we know that ${i_1},{i_2}$ are the currents carried and $d$ is the distance between the conductor. If both ${i_1},{i_2}$ are in the same direction then the conductors attract each other. Now while comparing magnetic fields created by conductor 1 at the location of conductor 2 and conductor 2 gets attracted towards conductor 1.

Then the force per unit le length between two thin long conductors carrying current is given by
$F = \dfrac{{{\mu _0}{i^2}}}{{2\pi d}}$
Therefore the thin long conductors carrying current are attracted to each other.

Hence, the correct answer is option (B).

Note: Biot-Savart’s Law for the magnetic field has certain similarities as well as differences with the Coulomb's law. In Biot-Savart’s Law the magnetic field is directly proportional to the sine of the angle. The magnetic field produced due to a current carrying segment is given by Biot Savart law. Biot-Savart’s Law is a vector quantity.