Problem 13 - University Physics with Modern 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: Identify relevant formulas

To solve the problem, we need two key formulas: the formula for the energy stored in an inductor, \( E = \frac{1}{2} L I^2 \), and the formula for the inductance of a toroidal solenoid, \( L = \frac{\mu_0 N^2 A}{2\pi r} \), 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, \( 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \).

Step 2: Rearrange the energy formula for inductance

Rearrange the energy formula to solve for inductance \( L \):\[ L = \frac{2E}{I^2} \]Plug in the given values (\( E = 0.390 \text{ J} \) and \( I = 12.0 \text{ A} \)) to find \( L \).

Step 3: Calculate the inductance

Substitute the given values to calculate inductance:\[ L = \frac{2 \times 0.390}{12.0^2} = \frac{0.780}{144} = 0.00542 \text{ H} \]

Step 4: Rearrange inductance formula to solve for turns

Using the inductance formula, rearrange it to solve for the number of turns \( N \): \[ N^2 = \frac{L \cdot 2\pi r}{\mu_0 A} \]Plug the calculated \( L \), given \( r = 0.15 \text{ m} \), \( A = 5.00 \times 10^{-4} \text{ m}^2 \), and \( \mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \) into the equation.

Step 5: Solve for the number of turns

Calculate \( N \):\[ N^2 = \frac{0.00542 \cdot 2\pi \cdot 0.15}{4\pi \times 10^{-7} \cdot 5.00 \times 10^{-4}} \]Simplify this to find:\[ N^2 = \frac{0.00542 \times 0.942}{2 \times 10^{-10}} = \frac{0.00510644}{2 \times 10^{-10}} = 25532.2 \]\[ N \approx \sqrt{25532.2} \approx 159.8 \]Thus, the number of turns \( N \approx 160 \).

Step 6: Finalize the result

The calculation shows that the number of turns for the toroidal solenoid is approximately 160 turns. Make sure to round to the nearest whole number as fractional turns are not feasible.

Key Concepts

Energy stored in an inductor

When you run an electric current through an inductor, such as a coil, energy is stored in its magnetic field. This stored energy can be useful in various applications, such as in transformers or inductors in circuits where energy needs to be temporarily stored. Understanding how this energy is stored and calculated is crucial, especially when working with components like solenoids or coils.
To find out how much energy is stored in an inductor, you can use the formula:

• \[ E = \frac{1}{2} L I^2 \]

Where:

• \( E \) is the energy stored (in joules),
• \( L \) is the inductance of the coil (in henries),
• \( I \) is the current flowing through the coil (in amperes).

The more current you pump through your inductor and the greater the inductance, the more energy gets stored.
For a given problem, if you already know the energy stored and the current used, you can rearrange this equation to solve for the inductance \( L \):

• \[ L = \frac{2E}{I^2} \]

By calculating \( L \), you lay the groundwork for further calculations, such as finding the number of turns in a coil.

Toroidal solenoid

A toroidal solenoid is a type of inductor with a coil shaped like a donut. This shape helps in efficiently containing the magnetic field inside its core, making it useful in various electronic applications like transformers and other electromagnetic devices. Because of its design, it minimizes the magnetic field exposure outside the coil, thus reducing electromagnetic interference.
To calculate the inductance of a toroidal solenoid, you use the formula:

• \[ L = \frac{\mu_0 N^2 A}{2\pi r} \]

Where:

• \( \mu_0 \) is the permeability of free space ( \( 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \) ),
• \( N \) is the number of turns,
• \( A \) is the cross-sectional area of the core (in square meters),
• \( r \) is the mean radius of the coil (in meters).

This formula shows that the inductance depends on the coil's physical characteristics: the number of turns, its size, and the area through which the magnetic field passes. The inherent nature of a toroidal solenoid always attempts to balance these variables to maintain efficiency and functionality.

Number of turns calculation

Calculating the number of turns in a coil helps determine how the inductor will perform. In our problem, once the inductance \( L \) is known, you need to solve for the number of turns \( N \) using the equation for inductance:

• \[ N^2 = \frac{L \cdot 2\pi r}{\mu_0 A} \]

To find \( N \) from this equation, follow these steps:

• Calculate the right side of the equation, which involves substituting known values for \( L \), \( r \), \( A \), and \( \mu_0 \).
• Take the square root of the result to solve for \( N \).

You should always round your answer to the nearest whole number. This is because a coil can't have a fraction of a turn; it needs to be a whole number for practical construction.
After these calculations, you have the number of turns, and thus one of the key variables required to analyze or design circuits involving your toroidal solenoid. Understanding these basics equips you with the knowledge to apply similar calculations to other electromagnetic devices.