Problem 48 - Physics

Question

The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of \(7.0 \mathrm{~T},\) and the current in the solenoid is \(2.0 \times 10^{2} \mathrm{~A} .\) What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.

Step-by-Step Solution

Answer

The number of turns per meter of the solenoid is approximately 28,000.

Step 1: Understand the Formula

The magnetic field strength inside a solenoid is given by the formula: \( B = \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \, \text{T·m/A}) \), \( n \) is the number of turns per unit length of the solenoid, and \( I \) is the current.

Step 2: Rearrange the Formula

We need to solve for \( n \), the number of turns per meter. Rearrange the formula to \( n = \frac{B}{\mu_0 \cdot I} \).

Step 3: Plug in the Known Values

Substitute the given values into the formula. The magnetic field \( B \) is \( 7.0 \, \text{T} \), the current \( I \) is \( 2.0 \times 10^{2} \, \text{A} \), and \( \mu_0 \) is \( 4\pi \times 10^{-7} \, \text{T·m/A} \). So, \( n = \frac{7.0}{4\pi \times 10^{-7} \times 200} \).

Step 4: Calculate the Result

Calculate the number of turns per meter. First, calculate the denominator: \( 4\pi \times 10^{-7} \times 200 = 2.513 \times 10^{-4} \). Then divide: \( n = \frac{7.0}{2.513 \times 10^{-4}} \approx 27836.5 \).

Step 5: Round Off

Round the result to a reasonable number of significant figures. Since the magnetic field and current are given to two significant figures, the number of turns per meter is \( n \approx 28000 \).

Key Concepts

Magnetic Field

A magnetic field is an invisible force field that surrounds magnetic materials and electric currents. It is a fundamental concept in physics, providing the force that attracts or repels magnetic objects. In the context of a solenoid, the magnetic field is created when an electric current passes through the coiled wire. The strength and direction of this field are determined by:

• the amount of current (I) flowing through the solenoid,
• the number of coils or turns of the wire, and
• the geometrical arrangement of these turns.

For solenoids, the magnetic field inside is uniform and its strength (B) can be calculated using the formula: \( B = \mu_0 \cdot n \cdot I \),where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per meter, and \( I \) is the current in amperes.
Understanding the magnetic field helps comprehend how devices like MRI machines operate by manipulating magnetic fields for imaging purposes.

MRI System

Magnetic Resonance Imaging (MRI) systems use strong magnetic fields and radio waves to create detailed images of the organs and tissues within the body. At the core of an MRI is a strong solenoid which acts as a giant magnet:

• The powerful magnetic fields align hydrogen atoms in the body.
• These atoms are then knocked out of alignment by a pulse of radio waves.
• As the atoms return to their original alignment, they emit signals.
• These signals are used to construct images of the body’s internal structures.

An MRI solenoid's design is crucial for the system's accuracy and performance. The solenoid must generate a strong and uniform magnetic field. In some MRI systems, this brings about the need for high turns per meter ratio in the solenoid to produce the required field strength, ensuring high-quality imaging.

Turns per Meter

The concept of turns per meter, often represented by the variable \( n \), refers to the number of loops or coils of wire present in one meter of the solenoid. This measure directly influences:

• The strength of the magnetic field inside the solenoid.
• The effectiveness of devices that rely on precise magnetic fields, such as MRI systems.

To calculate the turns per meter in a solenoid, you can use the formula:\[ n = \frac{B}{\mu_0 \cdot I} \]where \( B \) is the magnetic field strength, \( \mu_0 \) is the permeability of free space, and \( I \) is the current. Knowing the turns per meter helps in designing solenoids that meet specific field strength requirements in various applications.

Permeability of Free Space

The permeability of free space, denoted by \( \mu_0 \), is a constant used when calculating magnetic fields in a vacuum. It links the magnetic field in a solenoid to the electric current and number of turns per meter. Its value is:\( 4\pi \times 10^{-7} \text{ T·m/A} \).
This constant is crucial in many electromagnetic calculations because it defines how a magnetic field interacts within a vacuum. In the context of solenoids and MRI systems, it helps determine how effectively electric currents create magnetic fields. By exploring \( \mu_0 \), students can understand why different materials show different magnetic properties and how these properties influence engineering and physics designs.