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 conductors carrying current is calculated using the formula:
F = (μ₀ I₁ I₂ l) / (2 π r)
Where:
F: Force per unit length
μ₀: Permeability of free space = \(4\pi \times 10^{-7}\, \text{T.m/A}\)
I₁: Current in the first conductor
I₂: Current in the second conductor
l: Length of the conductor segment
r: Distance between the conductors
Step 2: Force on the Loop
For a square loop in a magnetic field:
Sides MN and NP will experience magnetic forces due to the magnetic field from the wire.
Sides MK and KP will not experience any net force because their magnetic fields are oriented to cancel each other out.
Step 3: Net Force on the Loop
The total force on the square loop is the sum of the forces on sides MN and NP. These forces act in opposite directions, resulting in a net force that depends on the distance from the wire.
Final Answer:
The magnitude of the magnetic force between the wire and the loop can be determined by substituting the given values into the force formula.
