Problem 6 - University Physics with Modern Physics

Question

A toroidal solenoid with mean radius \(r\) and cross-sectional area \(A\) is wound uniformly with \(N_{1}\) turns. A second toroidal solenoid with \(N_{2}\) turns is wound uniformly on top of the first, so that the two solenoids have the same cross-sectional area and mean radius. (a) What is the mutual inductance of the two solenoids? Assume that the magnetic field of the first solenoid is uniform across the cross section of the two solenoids. (b) If \(N_{1}=500\) turns, \(N_{2}=300\) turns, \(r=10.0 \mathrm{cm},\) and \(A=0.800 \mathrm{cm}^{2},\) what is the value of the mutual inductance?

Step-by-Step Solution

Answer

The mutual inductance is 24 \, \mu \text{H}.

Step 1: Define Mutual Inductance Formula

Mutual inductance (M) between two solenoids is given by:\[ M = \frac{\mu_0 N_1 N_2 A}{2 \pi r} \]where \( \mu_0 \) is the permeability of free space, \( N_1 \) and \( N_2 \) are the number of turns in each solenoid, \( A \) is the cross-sectional area, and \( r \) is the mean radius.

Step 2: Substitute Given Values

Substitute \( N_1 = 500 \) turns, \( N_2 = 300 \) turns, \( r = 10.0 \, \text{cm} = 0.1 \, \text{m} \), and \( A = 0.800 \, \text{cm}^2 = 0.00008 \, \text{m}^2 \) into the formula. Then use the permeability of free space \( \mu_0 = 4 \pi \times 10^{-7} \, \text{T m/A} \).

Step 3: Calculate Mutual Inductance

Using the values in the formula:\[M = \frac{4 \pi \times 10^{-7} \times 500 \times 300 \times 0.00008}{2 \pi \times 0.1}\]which simplifies to:\[M = \frac{4 \times 300 \times 0.00008 \times 500}{2 \times 0.1} \times 10^{-7}\]Finally, calculate \( M \) by solving the above expression.

Step 4: Final Calculation Result

The mutual inductance calculates to:\[M = 2.4 \times 10^{-5} \, \text{H}\]Therefore, the mutual inductance is \( 24 \, \mu \text{H} \).

Key Concepts

Toroidal Solenoid

A toroidal solenoid is essentially a coil shaped like a doughnut, specifically designed to produce uniform magnetic fields within its interior. It has distinct properties that make it useful in electromagnetic applications, especially in circuit and magnetic field interactions.

• **Structure:** It consists of wire wound in a coil over a circular path, usually with a circular cross-section.
• **Properties:** The key advantage is that the magnetic field generated by toroidal solenoids is nearly entirely internal, minimizing external magnetic interference.
• **Application:** They are used in various inductive devices and transformers due to their efficiency in containing the magnetic field.

A toroidal solenoid's unique doughnut shape ensures that the magnetic field lines are concentrated within the coil, reducing energy losses to the outside environment.

Magnetic Field

The magnetic field generated by a toroidal solenoid is a vital concept in understanding mutual inductance.

• **Formation:** The field is created when a current passes through the coil's turns.
• **Direction:** It forms closed loops within the solenoid’s circular path, ensuring that it stays mostly inside.
• **Uniformity:** The central area of the toroid maintains a nearly constant magnetic field due to symmetrical winding and the closed-loop system.

The strength and uniformity of this magnetic field within a toroidal solenoid are pivotal because they influence the mutual induction between two closely wound solenoids.

Permeability of Free Space

Permeability of free space, denoted by the symbol \( \mu_0 \), measures how easily magnetic fields pass through the vacuum of space or airy environments with no material interference.

• **Value:** It is a fundamental constant and equals \( 4 \pi \times 10^{-7} \text{ T m/A} \).
• **Role in Inductance:** In mutual inductance calculations, \( \mu_0 \) plays an essential part because it relates the magnetic flux density to the magnetic field strength.
• **Relevance:** It helps quantify the efficiency of magnetic field formation in the absence of magnetic materials.

Understanding the permeability of free space is crucial for calculating mutual inductance as it influences how a magnetic field propagates in an open-air setting without immediate solid material interference.

Cross-Sectional Area

The cross-sectional area \( A \) of a toroidal solenoid is the area of the circle formed by a cut perpendicular to the coiled wire.

• **Importance:** It is a key parameter in computing inductance as it affects how much magnetic flux can be channeled through the solenoid.
• **Relation to Inductance:** In the mutual inductance formula, the cross-sectional area directly influences the magnitude of mutual inductance by dictating the field strength interaction between the coils.
• **Units:** Measured in square meters (\( \text{m}^2 \)), it needs to be converted from any other units (like \( \text{cm}^2 \)) for the calculation.

A larger cross-sectional area allows more magnetic field lines to pass through, enhancing the coupling and thereby increasing mutual inductance.