Question

Consider a clinical MRI (magnetic resonance imaging) superconducting mag. net has a diameter of \(1.00 \mathrm{~m}\) length of \(1.50 \mathrm{~m}\), and a uniform magnetic field of 3.00 T. Determine (a) the energy density of the magnetic field and (b) the total energy in the solenoid.

Short answer

Based on the given solution: (a) The energy density of the magnetic field is \(\frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3}\). (b) The total energy in the solenoid is \(\frac{27}{32 \times 10^{-7}} \mathrm{J}\).

Step-by-step solution

Calculate the energy density of the magnetic field

The energy density of a magnetic field is given by the formula \(u = \frac{B^2}{2 \mu_0}\), where \(B\) is the magnetic field and \(\mu_0\) is the permeability of free space. The permeability of free space is a constant, \(\mu_0 = 4 \pi × 10^{-7} \mathrm{T \cdot m/A}\). Using the given magnetic field value of \(B = 3.00 \mathrm{T}\), we can find the energy density: \(u = \frac{(3.00\mathrm{T})^2}{2 \times (4 \pi × 10^{-7} \mathrm{T \cdot m/A})} = \frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3}\)

Calculate the volume of the solenoid

The solenoid has a cylindrical shape with a diameter of \(1.00 \mathrm{m}\) and a length of \(1.50 \mathrm{m}\). To calculate its volume, we first need to find the radius (which is half of the diameter) and then use the formula for the volume of a cylinder, \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height (length). The radius is \(0.50 \mathrm{m}\), so the volume of the solenoid is: \(V = \pi (0.50 \mathrm{m})^2 (1.50 \mathrm{m}) = \frac{3}{4}\pi \mathrm{m^3}\)

Calculate the total energy in the solenoid

Now that we have the energy density and the volume of the solenoid, we can calculate the total energy stored in the solenoid by simply multiplying the energy density by the volume. \(E = u \times V = \frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3} \times \frac{3}{4}\pi \mathrm{m^3} = \frac{27}{32 \times 10^{-7}} \mathrm{J}\) Therefore, the total energy in the solenoid is \(\frac{27}{32 \times 10^{-7}} \mathrm{J}\).

Key concepts

Magnetic Field

The magnetic field is an invisible force field that surrounds a magnet or a current-carrying conductor. It's a fundamental aspect of electromagnetism and dictates how magnetic materials interact with each other. The strength and direction of a magnetic field are represented by magnet field lines. When you think of a magnet, imagine these field lines looping from the North pole to the South pole, determining the magnet's pulling ability and the region's magnetic influence. In the context of the MRI machine considered in the exercise, the magnetic field strength is given as 3.00 Tesla (T). Tesla is the unit of measurement indicating the field's strength, where 1 T equals one Newton per Ampere meter. A strong magnetic field is crucial in MRI to align atomic particles in the body, allowing for precise imaging.

Energy Density

The concept of energy density describes the amount of energy stored in a given volume of space. For magnetic fields, energy density (\(u\)) is calculated using the formula: \(u = \frac{B^2}{2 \mu_0}\), where \(B\) is the magnetic field strength, and \(\mu_0\) is the permeability of free space.Energy density gives us insight into how much energy can be concentrated in a particular region, which is vital for understanding the efficiency and functioning of devices like MRI machines. A higher energy density means that more energy is stored in a smaller space. In the exercise, by calculating the energy density of the magnetic field inside the MRI, we can appreciate how concentrated and potent the magnetic energy is within that space.

Solenoid

A solenoid is a type of electromagnet, consisting of a wire wound into a coil. When electric current flows through it, a magnetic field is generated inside the coil. This property makes solenoids very useful in applications requiring controlled magnetic fields, such as in MRI machines. In this exercise's context, the solenoid creates the strong and uniform magnetic field required for MRI imaging. The solenoid is often cylindrical, as described by the exercise dimensions of 1.00 m in diameter and 1.50 m in length, making it resemble a round tube or pipe. Understanding the solenoid's volume is crucial for determining the total magnetic energy stored, which is done by calculating its cylindrical volume using radius and height. This setup allows medical practitioners to visualize detailed internal images of the human body.

Permeability of Free Space

Permeability of free space, represented by \(\mu_0\), is a fundamental physical constant that characterizes the relationship between magnetic field strength and magnetic flux density in a vacuum. Its value is approximately \(4\pi \times 10^{-7} \mathrm{T\cdot m/A}\) (Tesla meter per Ampere), and it plays a critical role in electromagnetism equations, including calculating energy density.This constant reflects how much resistance a vacuum offers against the formation of a magnetic field. By using \(\mu_0\) in calculations, we can determine parameters like energy density in magnetic systems. In the MRI exercise, including \(\mu_0\) allows us to accurately calculate the energy density of the magnetic field, thereby understanding the power harnessed in the MRI solenoid for precise medical imaging.