How Do You Calculate Mutual Inductance Between a Filament and a Square Loop?

[beramodk]

Homework Statement

Find the mutual inductance in air between a long straight filament and a coplanar square loop, 10cm on a side, with its nearest edge parallel to and 10cm from the filament.

Homework Equations

What equations should be best in solving this problem?

The Attempt at a Solution

I used M = N (Phi) / I, found B (flux density) and then found the magnetic total flux Phi, and I got the answer of
13.86 nH, but it seems that this way is a bit convoluted, if it is right.

Thanks for any help

 

[anathema]

!
Thank you for posting your question about mutual inductance. In order to solve this problem, there are a few equations that could be used. One possible approach is to use the formula for the mutual inductance between two parallel conductors, which is:
M = (μ0 * N1 * N2 * A) / L
where μ0 is the permeability of free space, N1 and N2 are the number of turns in the two conductors, A is the area of the loop, and L is the distance between the two conductors.

In this case, the long straight filament can be treated as a single turn, so N1 = 1. The square loop has a length of 10cm and a width of 10cm, so its area is A = (10cm)^2 = 100cm^2. The distance between the filament and the loop is also 10cm, so L = 10cm. Plugging these values into the formula, we get:
M = (4π * 10^-7 * 1 * 1 * 100cm^2) / 10cm = 4π * 10^-5 H = 12.57 μH.

Another approach that could be used is to calculate the magnetic flux through the loop and then use the formula M = Φ / I, where Φ is the magnetic flux and I is the current in the filament. To do this, we can use the formula for the magnetic field of a long straight filament, which is:
B = (μ0 * I) / (2π * r)
where μ0 is the permeability of free space, I is the current in the filament, and r is the distance from the filament to the loop.

In this case, we can set up a coordinate system with the filament running along the x-axis and the loop centered at the origin. The nearest edge of the loop is then located at x = 10cm. Using this information, we can calculate the magnetic field at the center of the loop (x = 0):
B = (4π * 10^-7 * 1A) / (2π * 10cm) = 2π * 10^-6 T.

To find the total magnetic flux through the loop, we can integrate the magnetic field over the area of the loop:
Φ = ∫∫B * d

 

[m2003]

.
Mutual inductance in air is a measure of how much magnetic flux is linked between two circuits. In this case, it represents the amount of magnetic flux that is created by the long straight filament and is linked with the coplanar square loop. The equation for mutual inductance is M = NΦ/I, where N is the number of turns in the circuit, Φ is the magnetic flux, and I is the current.

In order to solve this problem, you can use the following steps:

1. Determine the number of turns in the circuit. In this case, the long straight filament has one turn and the coplanar square loop has four turns (since it is a square loop).

2. Calculate the magnetic flux Φ. This can be done by using the equation Φ = μ0 * I * L / 2π * ln(b/a), where μ0 is the permeability of free space (4π*10^-7), I is the current, L is the length of the filament, and a and b are the distances from the filament to the nearest and farthest edges of the loop, respectively.

3. Substitute the values into the equation for mutual inductance, M = NΦ/I, and solve for M. In this case, you will get a value of 13.86 nH, which is the same result you obtained.

Overall, your approach was correct and the result is accurate. However, you can simplify the calculation by using the equation for magnetic flux directly, instead of finding the magnetic flux density first. Keep in mind that the equation for mutual inductance in air may change if the circuit is not in air, so it is important to double check the equation for the specific medium being used.