Question

When a magnet in an MRI is abruptly shut down, the magnet is said to be quenched. Quenching can occur in as little as \(20.0 \mathrm{~s}\). Suppose a magnet with an initial field of \(1.20 \mathrm{~T}\) is quenched in \(20.0 \mathrm{~s},\) and the final field is approximately zero. Under these conditions, what is the average induced potential difference around a conducting loop of radius \(1.00 \mathrm{~cm}\) (about the size of a wedding ring) oriented perpendicular to the field?

Short answer

Answer: The average induced potential difference around the conducting loop is approximately \(1.9\times 10^{-5}\mathrm{~V}\).

Step-by-step solution

Calculate the change in magnetic field

Since we know the initial and final magnetic fields, we can find the change in magnetic field which is \(\Delta B = B_f - B_0\). Given, \(B_0 = 1.20~T\) and \(B_f = 0~T\), we have: \(\Delta B = 0~T - 1.20~T = -1.20~T\)

Calculate the area of the conducting loop

We are given the radius of the conducting loop \(r = 1.00~cm = 0.010~m\). We can find the area of the loop using the formula for the area of a circle \(A = \pi r^2\): \(A = \pi (0.010~m)^2 = 0.000314~m^2\)

Calculate the change in magnetic flux

Now we can find the change in magnetic flux \(\Delta\phi\) using the formula: \(\Delta\phi = A\Delta B\): \(\Delta\phi = (0.000314~m^2)(-1.20~T) = -0.000377~Tm^2\)

Apply Faraday's law and find the induced emf

Faraday's law states that the induced electromotive force (emf) is equal to the negative rate of change of the magnetic flux: \(emf = -\frac{\Delta\phi}{\Delta t}\), with \(\Delta t\) being the time taken for the change in the magnetic field, which is given as \(20.0~s\): \(emf = -\frac{-0.000377~Tm^2}{20.0~s} = 0.00001885~V\)

Round the result to the appropriate number of significant figures

Given the initial data, we should round our answer to two significant figures: \(emf \approx 1.9 \times 10^{-5}~V\) The average induced potential difference around the conducting loop is approximately \(1.9\times 10^{-5}\mathrm{~V}\).

Key concepts

Magnetic Field Change

Understanding magnetic field change is crucial when discussing phenomena such as MRI magnet quenching. In our daily lives, magnetic fields are mostly perceived as static, but in numerous applications, they are dynamic and can change over time. Considering our example of an MRI machine, the magnetic field is intentionally collapsed from an initial high value to zero during a quench; this is an extreme example of a rapid change in a magnetic field.

The change in the magnetic field (\f\(\fDelta B\f\)) is simply the difference between the final (\f\(B_f\f\)) and the initial (\f\(B_0\f\)) magnetic field strengths. In this scenario, the magnetic field goes from a strong, uniform field to essentially none almost instantaneously. This rapid transformation is what ultimately leads to the creation of an induced electromotive force (emf), which we can observe and measure in conducting materials within the field, such as the hypothetical conducting loop the size of a wedding ring in the example.

Faraday's Law of Induction

Faraday's law of induction is a foundational concept in electromagnetism that explains how a change in magnetic flux induces an electromotive force (\f\(emf\f\)). According to Faraday's law, the \f\(emf\f\) induced in a closed circuit is directly proportional to the rate of change of the magnetic flux (\f\(\fDelta\flat\f\)) through the loop.

The formula for Faraday's law is given by \f\(emf = -\frac{\fDelta\flat}{\fDelta t}\f\), where \f\(\frac{\fDelta\flat}{\fDelta t}\f\) is the rate of change of the magnetic flux over time (\f\(\fDelta t\f\)). The negative sign in Faraday's law is a consequence of Lenz's law, which states that the direction of the induced \f\(emf\f\) and hence the current if one exists, will act to oppose the change in flux that produced it. Hence, if the magnetic flux through a loop increases, the induced \f\(emf\f\) will generate a current whose own magnetic field opposes the increase. This law is what allows us to calculate the induced \f\(emf\f\) when the MRI magnet undergoes quenching.

Magnetic Flux

The concept of magnetic flux (\f\(\flat\f\)) is integral to understanding Faraday's law of induction and MRI magnet quenching. Magnetic flux is a measure of the amount of magnetic field passing through a given area. It is expressed as the product of the magnetic field strength (\f\(B\f\)) and the perpendicular area (\f\(A\f\)) it penetrates - in the equation \f\(\flat = B \times A\f\). This physical quantity is comparable to imagining the number of magnetic field lines passing through a loop.

In the context of our MRI quenching example, when the magnetic field is abruptly shut down, the magnetic flux through a conducting loop changes substantially. Specifically, we calculated a change in flux (\f\(\fDelta\flat\f\)) by multiplying the change in the field with the area of the loop. The sudden disappearance of the magnetic field equates to a negative change in the magnetic flux (\f\(\flat\f\)), which, as Faraday's Law describes, induces an \f\(emf\f\).

Induced Electromotive Force (emf)

When delving into the principles of magnetism and electromagnetism, the concept of induced electromotive force (emf) stands out, especially in the context of changing magnetic fields as seen in the MRI magnet quenching scenario. The induced \f\(emf\f\) is the voltage produced in a conducting loop when it is exposed to a changing magnetic field. This change could be due to the movement of the loop in a stationary magnetic field, the movement of a magnet relative to the loop, or in this case, the sudden change of the magnetic field strength in a stationary loop.

As we applied Faraday's law, we discovered that the induced \f\(emf\f\) is proportional to the negative change in magnetic flux per unit time. In the MRI example, this phenomenon explains why an electric potential appeared across the loop; it's the magnetic field collapsing that generates it. The calculations showed that as the magnetic field went from 1.20 T (Tesla) to 0 T in only 20 seconds, this rapid change created a small but measurable voltage in the conducting loop, as computed in the steps of the provided solution.