Problem 50 - University Physics with Modern Physics

Question

An ideal toroidal solenoid (see Example 28.10) has inner radius \(r_1 =\) 15.0 cm and outer radius \(r_2 =\) 18.0 cm. The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) 12.0 cm; (b) 16.0 cm; (c) 20.0 cm?

Step-by-Step Solution

Answer

The fields are 0 T at 12.0 cm, non-zero at 16.0 cm, and 0 T at 20.0 cm.

Step 1: Identify the relevant formula

The magnetic field inside a toroidal solenoid is given by the formula \( B = \frac{\mu_0 n I}{2 \pi r} \), where \( \mu_0 \) is the permeability of free space \( 4\pi \times 10^{-7} \text{ Tm/A} \), \( n \) is the number of turns per unit length, \( I \) is the current, and \( r \) is the radius from the center of the solenoid where you want to calculate the magnetic field.

Step 2: Calculate the turns per unit length (n)

The total number of turns is 250, and these are distributed over the radial difference \( r_2 - r_1 \). Therefore, the turns per unit length \( n \) is calculated as \( n = \frac{250}{2\pi(r_2 - r_1)} \), where \( r_2 = 0.18 \text{ m} \) and \( r_1 = 0.15 \text{ m} \).

Step 3: Calculate the magnetic field at 12.0 cm

Using the formula \( B = \frac{\mu_0 n I}{2 \pi r} \), substitute \( r = 0.12 \text{ m} \). Since 12.0 cm is outside the solenoid's radius, the magnetic field is zero.

Step 4: Calculate the magnetic field at 16.0 cm

As 16.0 cm (0.16 m) lies between the inner and outer radius, use \( B = \frac{\mu_0 n I}{2 \pi \times 0.16} \) to find \( B \). Substitute \( \mu_0 = 4\pi \times 10^{-7} \text{ Tm/A} \), \( n \) from Step 2, and \( I = 8.50 \text{ A} \). Compute \( B \).

Step 5: Calculate the magnetic field at 20.0 cm

For \( r = 0.20 \text{ m} \), 20.0 cm is outside the solenoid; hence, the magnetic field is zero.

Key Concepts

Magnetic Field Inside a Toroidal Solenoid

A toroidal solenoid is a coil shaped like a donut, and inside it, a magnetic field is generated when current flows through it. The strength of this magnetic field at a point inside the solenoid is calculated by using the formula: \[ B = \frac{\mu_0 n I}{2 \pi r} \]. Here, \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current flowing through the solenoid.
The variable \( r \) is the radial distance from the center to the point inside the solenoid where you wish to calculate the magnetic field. This formula helps in finding out how strong or weak the magnetic field is at a specific distance from the center of the toroid.
It is important to remember that the magnetic field inside an ideal toroidal solenoid is circular and retains its strength as long as it is within the solenoidal region (between inner and outer radius). When you move outside the radial area between the inner and outer radii, the magnetic field effectively drops to zero.

Understanding Turns per Unit Length (n)

The concept of turns per unit length, denoted by \( n \), is crucial in understanding how densely wound the coils are in a solenoid. For a toroidal solenoid, this is determined by the formula:\[ n = \frac{N}{2\pi(r_2 - r_1)} \] where \( N \) is the total number of turns, \( r_1 \) is the inner radius, and \( r_2 \) is the outer radius of the toroid.
This formula effectively tells us the number of coil turns that fit into each meter (or another length unit) of space along the radial distance of the toroid. The higher the value of \( n \), the more tightly packed the turns are, leading to a stronger magnetic field, as evident from the magnetic field formula.
Understanding \( n \) is fundamental in designing solenoids with desired field strengths, especially in engineering and scientific applications where precise control of magnetic fields is needed.

Permeability of Free Space (\( \mu_0 \))

Permeability of free space, symbolized by \( \mu_0 \), is a fundamental physical constant with the value \( 4\pi \times 10^{-7} \text{ Tm/A} \). This constant is crucial because it represents how much resistance the magnetic field encounters in a vacuum.
In the context of solenoids, \( \mu_0 \) plays a pivotal role in determining the magnetic field strength generated by a given solenoid. The parameter \( \mu_0 \) ensures that the units in the magnetic field formula work out correctly, linking the physical quantities of current, turns, and radius into a coherent measure of magnetic field strength.
Learning about \( \mu_0 \) gives insights into not just solenoid applications but also broader electromagnetic theories and phenomena.

What Makes a Solenoid 'Ideal'?

An ideal solenoid is a theoretical construct where the magnetic field inside the solenoid is uniform and perfectly confined, while the magnetic field outside is zero. This idealization assumes:

• Infinite length of the solenoid.
• Perfectly uniform coil winding.
• No leakage of the magnetic field outside the solenoid.

An ideal toroidal solenoid approximates this by bending the solenoid into a closed loop, which helps confine the magnetic field lines inside the coil loop.
Even though real solenoids can't reach this ideal due to physical constraints, the ideal model simplifies mathematical treatments and provides a good approximation for many practical scenarios.
Knowing about ideal solenoids helps in understanding real-world applications and limitations, guiding engineers and scientists in effective solenoid design for specific uses.