BIO One possible concern with MRI (see Exercise 28) is turning the magnetic field on or off too quickly. Bodily fluids are conductors, and a changing magnetic field could cause electric currents to flow through the patient. Suppose a typical patient has a maximum cross-section area of 0.060 m². What is the smallest time interval in which a 5.0 T magnetic field can be turned on or off if the induced emf around the patient's body must be kept to less than 0.10 V?

Verified step by step guidance

1. Understand the problem: The goal is to determine the smallest time interval required to turn a magnetic field on or off such that the induced emf (electromotive force) around the patient's body is less than 0.10 V. This involves Faraday's Law of Induction, which relates the rate of change of magnetic flux to the induced emf.
2. Recall Faraday's Law of Induction: The induced emf (ε) is given by the formula: $\epsilon =-\frac{d\Phi }{}$_mdt, where $\Phi$_m is the magnetic flux. Magnetic flux is defined as $\Phi$_m=BAcosθ, where $B$ is the magnetic field strength, $A$ is the area, and $\theta$ is the angle between the magnetic field and the normal to the area. Assume $\theta =0$ (field perpendicular to the area) for maximum flux.
3. Substitute the given values into the flux formula: The magnetic flux is $\Phi$_m=BA. Here, $B=5.0$ T and $A=0.060$ m². Calculate the initial flux value.
4. Rearrange Faraday's Law to solve for the time interval: From $\epsilon =-\frac{d\Phi }{}$_mdt, we can write $dt=\frac{d\Phi }{}$_mε. Substitute $\epsilon =0.10$ V and the change in flux $d\Phi$_m=5.0×0.060 T·m².
5. Perform the division to find the smallest time interval: Use the formula $dt=\frac{d\Phi }{}$_mε to calculate the time interval. Ensure the units are consistent (T·m² for flux and V for emf). This will give the smallest time interval required to keep the induced emf below 0.10 V.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (emf) in the loop. The induced emf is directly proportional to the rate of change of the magnetic flux through the loop. This principle is crucial for understanding how electric currents can be generated in conductors, such as bodily fluids, when exposed to varying magnetic fields.

Magnetic Flux

Magnetic flux refers to the total magnetic field passing through a given area, calculated as the product of the magnetic field strength and the area perpendicular to the field. It is measured in webers (Wb). In the context of the question, the change in magnetic flux as the magnetic field is turned on or off is what induces the emf in the patient's body.

Induced EMF and Time Interval

The induced emf can be expressed as the negative rate of change of magnetic flux over time, as described by Faraday's Law. To keep the induced emf below a specified threshold, the time interval for changing the magnetic field must be calculated. This relationship highlights the trade-off between the strength of the magnetic field and the speed at which it can be altered without exceeding safety limits.