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.
Show Hint
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.
Step 1: Magnetic Force Between Two Parallel Conductors
The force per unit length between two parallel current-carrying conductors is given by:
F = (μ₀ I₁ I₂ l) / (2 π r)
Where:
F: Force per unit length between the wire and the loop
μ₀: Permeability of free space = \(4\pi \times 10^{-7}\, \text{T.m/A}\)
I₁: Current in the wire
I₂: Current in the loop
l: Side length of the square loop
r: Distance between the wire and the loop
Step 2: Force on the Loop
For the square loop:
The magnetic force will act on the sides MN and NP due to the magnetic field generated by the current in the wire.
The other two sides, MK and KP, will not experience any force due to the orientation of the magnetic field
Step 3: Net Force on the Loop
The net force on the square loop is the sum of the forces on the opposite sides MN and NP. These forces will act in opposite directions, leading to a net force based on the distance between the wire and the loop.
Final Answer:
The magnitude of the magnetic force exerted by the wire on the loop can be calculated by substituting the known values into the force formula.
