Question

Two toroidal solenoids are wound around the same form so that the magnetic field of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a) What is the mutual inductance of the pair of solenoids? (b) When the current in solenoid 2 is 2.54 A, what is the average flux through each turn of solenoid 1?

Short answer

(a) 0.00491 H (b) 0.0125 Wb

Step-by-step solution

Calculate Mutual Inductance

To find the mutual inductance \( M \), we start with the formula that relates the mutual inductance to the magnetic flux: \( \Phi_{21} = M I_1 \). Here, \( \Phi_{21} = 0.0320 \ \text{Wb} \) (the average flux through each turn of solenoid 2 due to solenoid 1), and \( I_1 = 6.52 \ \text{A} \). Thus, \( M = \frac{\Phi_{21}}{I_1} = \frac{0.0320}{6.52} \approx 0.00491 \ \text{H} \).

Calculate Average Flux through Solenoid 1

To find the average flux through each turn of solenoid 1 when the current in solenoid 2 is given, we use the mutual inductance calculated before: \( \Phi_{12} = M I_2 \). Here, \( M = 0.00491 \ \text{H} \) and \( I_2 = 2.54 \ \text{A} \). So, \( \Phi_{12} = 0.00491 \times 2.54 = 0.0125 \ \text{Wb} \).

Key concepts

Magnetic Flux

Magnetic flux is an essential concept when discussing mutual inductance between two solenoids. It provides a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. Magnetic flux is represented by the Greek letter \( \Phi \) and is measured in webers (Wb).

The relationship between magnetic flux and the magnetic field can be seen in the expression \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the field lines and the normal to the area. By understanding this equation, one can appreciate how changes in the magnetic field or the area influence the magnetic flux, which in turn is vital for understanding mutual inductance.

Solenoids

A solenoid, at its core, is a coil of wire designed to harness electricity to produce a magnetic field. When electric current flows through the solenoid, it generates a magnetic field inside it. This magnetic field can induce an electromotive force or flux in nearby coils, showcasing the principle of mutual inductance.

In the mutual inductance exercise example, two toroidal solenoids were considered, each affecting the other through the magnetic field they produce. The parameters of a solenoid, such as the number of turns, the current flowing through it, and the coil's geometry, all play crucial roles in determining the strength and impact of the produced magnetic field on other nearby solenoids.

Toroidal Solenoid

A toroidal solenoid is a special kind of solenoid that is formed into the shape of a donut or a torus. The design of the toroidal solenoid is beneficial because it makes the magnetic field lines confined mostly within the core of the solenoid, creating a very efficient magnetic field.

This confinement means there's little leakage of the magnetic field outside the coil, which minimizes interference with other components in a circuit. The toroidal solenoid setup utilized in the exercise ensures that the magnetic field lines produced by one solenoid effectively pass through the turns of the other, facilitating a strong interaction and mutual inductance.

Inductance Calculation

Inductance is a measure of the capability of a coil to store energy in a magnetic field. In particular, mutual inductance describes how a change in current in one coil induces an electromotive force in a second, nearby coil.

To calculate the mutual inductance between two solenoids, we used the formula \( M = \frac{\Phi_{21}}{I_1} \), where \( \Phi_{21} \) is the average magnetic flux through one solenoid due to the current in the other. By knowing the current flowing through solenoid 1 and the resultant flux in solenoid 2, we determined the mutual inductance as approximately 0.00491 Henries (H).

• This calculation highlights the direct proportionality between flux and current, within the solenoid configurations.
• Understanding mutual inductance aids in designing efficient transformers and circuits that rely on magnetic coupling between coils.

Proper calculation and understanding of mutual inductance are essential in various applications, from power systems to sensor technologies.