There are two circular loop of wires and each has a single turn. One loop has radius R and the other 2R radius. The is a counterclockwise current flows in both loops. These two loops are placed on top of each other. Hence the smaller loop is inside the larger loop. That means both loop have the same center point. What is the magnitude of the magnetic field at the center point?
To find the magnitude of the magnetic field at the center point, we can use Ampere's law.
Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space (μ₀).
Mathematically, it can be written as:
∮ B · dl = μ₀ * I_enc
Where ∮ B · dl represents the line integral of the magnetic field B around a closed loop, I_enc is the current passing through the loop enclosed by the closed loop, and μ₀ is the permeability of free space.
In this case, since the current is flowing counterclockwise in both loops and they share the same center, the smaller loop is completely enclosed by the larger loop. Therefore, we can consider the larger loop to be the closed loop for applying Ampere's law.
The larger loop has a radius of 2R, so its circumference is 2π(2R) = 4πR. The current passing through the loop is the same for both loops.
Using Ampere's law:
∮ B · dl = μ₀ * I
∮ B · dl = B(2πR)
B(2πR) = μ₀ * I
Solving for B, the magnitude of the magnetic field at the center point:
B = (μ₀ * I) / (2πR)
Since there is only one turn of wire in each loop, the current I is the same for both loops.
Therefore, the magnitude of the magnetic field at the center point is:
B = (μ₀ * I) / (2πR)