Problem 42 - College Physics

Question

An air-filled toroidal solenoid has a mean radius of 15.0 \(\mathrm{cm}\) and a cross-sectional area of 5.00 \(\mathrm{cm}^{2} .\) When the current is \(12.0 \mathrm{A},\) the energy stored is 0.390 \(\mathrm{J} .\) How many turns does the winding have?

Step-by-Step Solution

Answer

The toroidal solenoid has approximately 160 turns.

Step 1: Understanding the Formula for Inductance

The inductance (L) of a toroidal solenoid can be given by the formula: \( L = \frac{\mu_0 n^2 A}{2\pi r} \), where \(n\) is the number of turns, \(A\) is the cross-sectional area, \(r\) is the mean radius, and \(\mu_0\) is the permeability of free space: \(\mu_0 = 4\pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m/A}\).

Step 2: Relating Inductance to Energy Storage

The energy stored in an inductor \( E \) is given by: \( E = \frac{1}{2} L I^2 \), where \( I \) is the current. We can rearrange this formula to solve for \(L\), giving \( L = \frac{2E}{I^2} \).

Step 3: Calculate the Inductance

Substitute the given values into the energy formula to find the inductance: \( E = 0.390 \mathrm{~J} \) and \( I = 12.0 \mathrm{~A} \). Thus, \( L = \frac{2 \times 0.390}{12^2} = \frac{0.780}{144} = 0.005417 \mathrm{~H} \).

Step 4: Solve for Number of Turns

Rearrange the inductance formula from Step 1 to find the number of turns \(n\): \( n = \sqrt{\frac{2\pi r L}{\mu_0 A}} \). Substitute \( r = 0.15 \mathrm{~m} \), \( A = 5.00 \times 10^{-4} \mathrm{~m}^2 \), \( L = 0.005417 \mathrm{~H} \), and \( \mu_0 = 4\pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m/A}\).

Step 5: Calculate the Number of Turns

Plug the values into the formula for \( n \): \( n = \sqrt{\frac{2\pi \times 0.15 \times 0.005417}{4\pi \times 10^{-7} \times 5.00 \times 10^{-4}}} \). Simplifying inside the square root gives \( n = \sqrt{\frac{0.005087}{2 \times 10^{-10}}} \). Further simplification gives \( n \approx \sqrt{25435} \approx 159.5 \approx 160 \) (since the number of turns should be an integer).

Key Concepts

Understanding Inductance in a Toroidal Solenoid

Inductance in electrical circuits often refers to the property that allows a solenoid to store energy when electrical current flows through it. The toroidal solenoid, which is donut-shaped, helps concentrate the magnetic field, aligning it with the path of the wire. This is essential in maximizing the inductance.
Inductance (\(L\)) of a toroidal solenoid can be calculated with the formula \(L = \frac{\mu_0 n^2 A}{2\pi r}\). Here:

• \(\mu_0\) is the permeability of free space, a constant that represents how much resistance the vacuum gives to forming a magnetic field, valued at \(4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}\).
• \(n\) is the number of turns in the solenoid, which affects the inductance directly through squaring. More turns mean greater inductance.
• \(A\) is the cross-sectional area of the donut, influencing how much magnetic field is produced.
• \(r\) is the mean radius of the solenoid, contributing to how the magnetic field wraps inside the solenoid.

How Energy Storage Works in a Solenoid

Energy storage in a solenoid is similar to a spring storing potential energy. When electric current flows through the coil of the solenoid, it generates a magnetic field, storing energy in this field.
The formula connecting inductance to the energy stored in a magnetic field is \(E = \frac{1}{2} L I^2\).

• Here, \(E\) is the energy in joules stored in the solenoid.
• \(L\) is the inductance, knotted directly to the number of turns and the solenoid's properties.
• \(I\) is the current in amperes, squared in the equation showing the exponential relationship with energy; meaning even a small increase in current results in a larger increase in stored energy.

This energy storage principle is crucial in fields like power transfer and radio transmission, where energy conservation and storage play vital roles.

Importance of the Number of Turns

The number of turns (\(n\)) is a critical factor in defining the behavior and characteristics of a solenoid. As seen from the inductance formula, \(n\) appears square:

• This squaring effect means that even minor alterations in the number of turns can lead to significant changes in the inductance.
• The more turns a solenoid has, the stronger the magnetic field produced, because each loop contributes its share of magnetic field strength, compounding the effect.
• Manufacturers can adjust the solenoid's design by changing the number of loops to meet different requirements for inductance.

Designing a solenoid for specific needs often begins with determining the necessary \(n\) to meet desired inductance and energy storage goals. It's not just about fitting wires but about precision in meeting electrical and magnetic requirements.

Magnetic Field in Toroidal Solenoids

The magnetic field in a toroidal solenoid has unique characteristics due to its shape.
Unlike a regular solenoid, the magnetic field in a toroid is contained, looping around the core without leaking out.

• This self-contained field design makes it efficient, minimizing losses and interference from external magnetic fields. This is because the field does not extend beyond the solenoid, unlike the bar-shaped solenoids.
• A very concentrated field helps in applications where precise control of magnetic interactions is required, such as in transformers and inductors.
• The efficiency of this structure is also evaluated through parameters like the mean radius (\(r\)) and cross-sectional area (\(A\)). Together, they impact how the magnetic path is shaped within the toroid.

The containment of magnetic fields in toroidal solenoids not only optimizes inductance but also helps in situations where space and energy efficiency are crucial, like in electromagnetic sensors.