A square loop of side "a" carries current "I". It creates a magnetic field "B-square" at its center. The same loop is then reshaped into a circle, which lowers the magnetic field "B-circle" at the center. What is the ratio B-square/B-circle?
circle:
L=2πR
R=L/2π
B₀=μ₀I/2R= μ₀I2 π /2L=
=μ₀πI/L
square:
L=4a=> a=L/4
The distance from the center of the square to the midpoint of the side of the square is
b=a/2tan45=a/2
B⋄=4B₁=4B=
=4(μ₀/4π)I/b∫cosφdφ {limits: (- π/4) (+π/4)}
=8(μ₀/4π)I/a•√2=2√2 μ₀I/πa=
=8√2 μ₀I/πL
B⋄/B₀={8√2 μ₀I/πL}/{ μ₀πI/L}=8√2/π²=1.15
To find the ratio of the magnetic field B-square created by the square loop to the magnetic field B-circle created by the reshaped loop, we can use the formula for the magnetic field at the center of a current loop.
For a square loop of side "a" carrying current "I", the magnetic field B-square at its center can be calculated using the formula:
B-square = (μ₀ * I) / (2 * a)
where μ₀ is the permeability of free space.
When the square loop is reshaped into a circle, the radius of the circle will be the same as half the length of the side of the square (r = a/2).
For a circular loop with radius "r" carrying the same current "I", the magnetic field B-circle at its center can be calculated using the formula:
B-circle = (μ₀ * I) / (2 * r)
Substituting r = a/2 into the formula, we get:
B-circle = (μ₀ * I) / (2 * (a/2)) = (μ₀ * I) / a
To find the ratio B-square/B-circle, we can divide the two equations:
(B-square / B-circle) = [(μ₀ * I) / (2 * a)] / [(μ₀ * I) / a]
Simplifying this expression, we find:
(B-square / B-circle) = (a/2) / a = 1/2
Therefore, the ratio of B-square to B-circle is 1/2.
To find the ratio of magnetic fields, we need to compare the magnetic field at the center of the square loop with the magnetic field at the center of the reshaped circle.
Let's start by calculating the magnetic field at the center of the square loop. The magnetic field produced at the center of a square loop can be calculated using Ampere's law. Ampere's law states that the magnetic field around a closed loop is proportional to the current enclosed by the loop.
In this case, the square loop carries current I. The magnetic field produced at the center of the square loop, B-square, can be calculated using the formula:
B-square = μ₀ * (I / (2√2a))
Where:
μ₀ is the permeability of free space (constant)
I is the current through the loop
a is the length of one of the sides of the loop
Next, let's consider the reshaped circle. Since the loop is reshaped into a circle, the current will be uniformly distributed along the circumference of the circle. To find the magnetic field at the center of the circle, B-circle, we can make use of the Biot-Savart law. The Biot-Savart law states that the magnetic field at a point due to a current element is directly proportional to the current through that element and inversely proportional to the square of the distance between the element and the point.
For a circular loop, the magnetic field at the center is given by:
B-circle = μ₀ * (I / (2R))
Where:
μ₀ is the permeability of free space (constant)
I is the current through the loop (distributed uniformly along the circumference)
R is the radius of the circle
Now we have the expressions for both B-square and B-circle. To find the ratio B-square/B-circle, we can substitute the expressions into the ratio:
B-square/B-circle = (μ₀ * (I / (2√2a))) / (μ₀ * (I / (2R)))
The permeability of free space (μ₀) cancels out, leaving us with:
B-square/B-circle = R / (√2a)
Hence, the ratio of the magnetic fields, B-square/B-circle, is equal to R divided by the square root of 2 times the length of one side of the square loop.