A square loop MNPK of side \( l \) carrying a current \( I_2 \) is kept close to a long straight wire in the same plane and the wire carries a steady current \( I_1 \) as shown in the figure. Obtain the magnitude of magnetic force exerted by the wire on the loop.
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The magnetic force between a current-carrying wire and a loop depends on the current, the distance between the wire and the loop, and the length of the loop. The force on the loop is a result of the varying magnetic field strength along the loop's sides.
The force between a current-carrying wire and a magnetic field is given by the formula:
\[
F = I L B \sin \theta
\]
where:
- \( I \) is the current in the wire,
- \( L \) is the length of the wire segment in the magnetic field,
- \( B \) is the magnetic field strength, and
- \( \theta \) is the angle between the direction of current and the magnetic field.
Step 1: Magnetic field due to the long straight wire
The magnetic field at a distance \( r \) from a long straight current-carrying wire is given by:
\[
B = \frac{\mu_0 I_1}{2 \pi r}
\]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I_1 \) is the current in the long straight wire,
- \( r \) is the distance from the wire (which in this case is the distance from the wire to the side of the square loop).
Step 2: Magnetic force on each segment of the square loop
The force on a segment of the loop is due to the magnetic field produced by the wire. The force on a current-carrying segment in a magnetic field is:
\[
F = I_2 L B
\]
For each side of the square loop, the magnetic field will have different magnitudes based on the distance from the wire, but the force will act along the direction of the current in each segment.
Step 3: Net Force on the loop
The total force on the loop can be found by considering the contribution from all four sides of the square loop. The forces on the opposite sides of the square will be equal in magnitude but opposite in direction. The net force on the square loop will be the sum of these individual forces, which depends on the distance from the wire and the current in the wire.
The force on the segments will not cancel out because they are at different distances from the wire. To calculate the total force, we integrate the force over each side of the loop, taking into account the varying magnetic field strength as the distance from the wire changes.
Final Expression for Force
After performing the integration, the total magnetic force exerted by the wire on the loop is given by:
\[
F = \frac{\mu_0 I_1 I_2 l}{2 \pi} \left( \frac{1}{d} - \frac{1}{d + l} \right)
\]
where:
- \( d \) is the initial distance of the square loop from the wire,
- \( l \) is the side length of the square loop.
This formula gives the magnitude of the net magnetic force exerted by the current-carrying wire on the square loop.
