Question

The magnetic field existing in a region is given by  `vecB = B_0(1 + x/1)veck` . A square loop of edge l and carrying a current i, is placed with its edges parallel to the x−y axes. Find the magnitude of the net magnetic force experienced by the loop.

Answer 1

Given:
Magnetic field, `vecB = B_0(1 + x/1)veck`
Length of the edge of a square loop = l
Electric current flowing through it = i
As per the question, the loop is placed with its edges parallel to the X−Y axes.

In the figure, arrow denotes the direction of force on different sides of the square.
The net magnetic force experienced by the loop,
`vecF = ivecl  xx vecB`
Force on AB:
Consider a small element of length dx at a distance x from the origin on line AB.
Force on this small element,
dF = iB_0 on the full length of AB,
F_AB = \[\int\limits_{x=0}^{x=0}\] iB_0 `(1 + x/l)`

      = `iB_0` \[\int\limits_{x=0}^{x=0}\] `(dx + 1/l xdx)`
      = `iB_0(l + 1/2)`

      = `(3iBgl)/(2)`
Force on AB will be acting downwards.
Similarly, force on CD,
`F_2 = iB_0 (l + l/2)`

       `=(3iBgl)/(2)`
Force on AB will be acting downwards.
Similarly, force on CD,
`F_2 = iB_0 (l + 1/2)`
      = `(3iBgl)/2`
So, the net vertical force = F₁ − F₂ = 0
Force on AD,
`F_4 = iB_0l (1 + 1/l)`
       = 2iB₀l
Force on BC
`F_4 = iB_0l(1 + 1/l)`
=2iB₀l
So, the net horizontal force = F₄−F₃ = iB₀l