Problem 12 - University Physics with Modern Physics

Question

In many magnetic resonance imaging (MRI) systems, the magnetic field is produced by a superconducting magnet that must be kept cooled below the superconducting transition temperature. If the cryogenic cooling system fails, the magnet coils may lose their superconductivity and the strength of the magnetic field will rapidly decrease, or \(quench\). The dissipation of energy as heat in the now-nonsuperconducting magnet coils can cause a rapid boil-off of the cryogenic liquid (usually liquid helium) that is used for cooling. Consider a superconducting MRI magnet for which the magnetic field decreases from 8.0 T to nearly 0 in 20 s. What is the average emf induced in a circular wedding ring of diameter 2.2 cm if the ring is at the center of the MRI magnet coils and the original magnetic field is perpendicular to the plane that is encircled by the ring?

Step-by-Step Solution

Answer

The average induced emf is approximately 0.000152 V.

Step 1: Understanding the Problem

We are asked to find the average electromotive force (emf) induced in a circular ring when the magnetic field through it changes over time. The magnetic field decreases from 8.0 T to nearly 0 within 20 seconds, and the diameter of the ring is given as 2.2 cm.

Step 2: Calculate the Area of the Ring

The area of a circle, which is the area encircled by the ring, is given by the formula \( A = \pi r^2 \). First, calculate the radius from the diameter: \( r = \frac{2.2}{2} = 1.1 \) cm, which is 0.011 m. Thus, the area is \( A = \pi (0.011)^2 \approx 3.80 \times 10^{-4} \text{ m}^2 \).

Step 3: Calculate the Change in Magnetic Flux

Magnetic flux \( \Phi \) through the ring is given by \( \Phi = B \cdot A \), where \( B \) is the magnetic field, and \( A \) is the area calculated. Initially, \( \Phi_i = 8.0 \times 3.80 \times 10^{-4} = 3.04 \times 10^{-3} \) Webers. Finally, when the magnetic field decreases to nearly 0, \( \Phi_f = 0 \). Thus, the change in magnetic flux \( \Delta \Phi = \Phi_f - \Phi_i = -3.04 \times 10^{-3} \) Webers.

Step 4: Calculate the Average EMF

The average emf \( \mathcal{E} \) induced in the ring can be calculated using Faraday's law of electromagnetic induction: \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \). Substituting the values, \( \Delta t = 20 \) seconds, we get \( \mathcal{E} = -\frac{-3.04 \times 10^{-3}}{20} \approx 1.52 \times 10^{-4} \) Volts.

Key Concepts

Magnetic Field

A magnetic field is a fundamental aspect of electromagnetism. It describes a field around a magnetic material or a moving electric charge. This field exerts a force on other nearby moving charges and magnets. In the context of an MRI machine, the magnetic field is crucial as it aligns protons in the body, which are then used to produce images. Magnetic fields are often measured in teslas (T), a unit that signifies the strength of the magnetic field. For instance, the original field strength of 8.0 T in our problem is quite strong compared to the Earth's magnetic field, which is about 0.00005 T.
Magnetic fields can vary in strength and direction. A consistent field allows for stable operation of equipment like MRI machines. However, when issues arise, such as cooling system failures, the resulting change can significantly impact the system's effectiveness. This change, or quench, leads to a rapid decline in the magnetic field strength, as seen in the exercise where it falls to nearly zero from 8.0 T.

Faraday's Law

Faraday's law of electromagnetic induction is a cornerstone principle in electromagnetism. It explains how a change in magnetic field within a closed loop induces an electromotive force (emf). The law can be mathematically expressed through the formula: \[\mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \]
Where \(\mathcal{E}\) is the induced emf, \(\Delta \Phi\) is the change in magnetic flux, and \(\Delta t\) is the time over which this change occurs. In the context of our problem, a changing magnetic field in the MRI can induce an emf in objects nearby, such as the wedding ring.
The negative sign in Faraday's law indicates Lenz's Law, which states the induced emf will oppose the change in flux. This concept is important to understand how protective measures work in systems using strong magnetic fields. In our example, the ring at the center experiences a significant decrease in magnetic flux, from 3.04 x 10^{-3} Webers to nearly zero, which results in the calculated emf of 1.52 x 10^{-4} Volts.

Superconductivity

Superconductivity is a fascinating property of certain materials where they exhibit zero electrical resistance and expulsion of magnetic fields when cooled below a characteristic critical temperature. This phenomenon allows for perfect conductivity, enabling magnetic resonance imaging (MRI) systems to generate powerful magnetic fields efficiently. This state requires materials to be extremely cold, often involving the use of cryogenic liquids like liquid helium.
When the cooling system in an MRI fails, the superconducting state can be lost, a situation known as a quench. During a quench, the material transitions back into a normal resistive state, resulting in the loss of superconductivity. This change causes the magnetic field to rapidly collapse, as seen in the problem where the field decreases from 8.0 T to near zero in seconds.

• In a superconducting state, magnetic fields are expelled, a property termed the Meissner effect.
• Superconductors can carry large currents, which are integral in generating high magnetic fields crucial for MRI.

Without the cooling, energy is no longer being conserved, and it dissipates as heat. This heat could further cause a boil-off of the cooling liquid, leading to equipment and safety concerns."