Mutual Inductance Between Square and Circle Circuit

[~Sam~]

Homework Statement

There are two circuits in the XY plane: one is a square (side 0.2 m in
length) centered on the origin. The second is circle or radius r also
centered on the origin. The circle is smaller than fits inside) the square.

By assuming the radius of the circle is small compared to
the length of the sides of the square, describe how you would estimate the mutual inductance between the two circuits. Also, describe how you would determine the magnetic field Bz due to the square circuit.

Homework Equations

There are several equations, but I'm not sure all apply..these include the Biot-Savart law, Neumann formula, etc.

The Attempt at a Solution

Okay so how I how to deal with the inductance of other basic setups like 2 loops etc. But I don't know how to go by a small circle circuit with a square. Any ideas?

 

[mark726]

To estimate the mutual inductance between the two circuits, you can use the Neumann formula, which states that the mutual inductance (M) between two circuits is equal to the product of the current in one circuit (I1), the current in the other circuit (I2), and the integral of the magnetic field (B) over the area enclosed by the two circuits (A).

In this case, since the circle is smaller than the square, we can assume that the magnetic field in the area of overlap between the two circuits is approximately uniform. Therefore, we can simplify the Neumann formula to M = μ0 * I1 * I2 * A, where μ0 is the permeability of free space.

To determine the magnetic field Bz due to the square circuit, you can use the Biot-Savart law, which states that the magnetic field at a point (P) due to a current element (dI) at a distance (r) is equal to μ0/4π * dI * sin(θ)/r^2, where θ is the angle between the current element and the line connecting the current element to the point P.

In this case, since the square is centered on the origin, we can assume that the magnetic field at any point on the XY plane due to the square circuit is only in the z-direction (Bz). Therefore, we can simplify the Biot-Savart law to Bz = μ0/4π * I * (L/2)/r^2, where I is the current in the square circuit and L is the length of one side of the square.

To calculate the mutual inductance and magnetic field Bz, you will need to know the current in each circuit. This can be measured experimentally or calculated using the circuit's resistance and voltage. Once you have the current values, you can plug them into the equations above to estimate the mutual inductance and magnetic field Bz between the two circuits.