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Ch 30: Electromagnetic Induction
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 30: Electromagnetic InductionProblem 28a
Chapter 30, Problem 28a

BIO MRI (magnetic resonance imaging) is a medical technique that produces detailed 'pictures' of the interior of the body. The patient is placed into a solenoid that is 40 cm in diameter and 1.0 m long. A 100 A current creates a 5.0 T magnetic field inside the solenoid. To carry such a large current, the solenoid wires are cooled with liquid helium until they become superconducting (no electric resistance). How much magnetic energy is stored in the solenoid? Assume that the magnetic field is uniform within the solenoid and quickly drops to zero outside the solenoid.

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Step 1: Start by recalling the formula for the energy density of a magnetic field, which is given by: \( u = \frac{B^2}{2\mu_0} \), where \( B \) is the magnetic field strength and \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} \)). This formula gives the energy stored per unit volume in the magnetic field.
Step 2: Calculate the volume of the solenoid. The solenoid is cylindrical, so its volume \( V \) can be calculated using the formula for the volume of a cylinder: \( V = \pi r^2 l \), where \( r \) is the radius of the solenoid and \( l \) is its length. The diameter of the solenoid is given as 40 cm, so the radius \( r \) is half of that (20 cm or 0.2 m). The length \( l \) is given as 1.0 m.
Step 3: Multiply the energy density \( u \) by the volume \( V \) to find the total magnetic energy stored in the solenoid. The formula for the total magnetic energy is \( U = u \cdot V \), which can also be written as \( U = \frac{B^2}{2\mu_0} \cdot \pi r^2 l \). Substitute the values for \( B \), \( \mu_0 \), \( r \), and \( l \) into this formula.
Step 4: Simplify the expression by performing the necessary algebraic operations. Ensure that all units are consistent (e.g., meters for length and Tesla for magnetic field strength). This will give you the total magnetic energy stored in the solenoid in joules (J).
Step 5: Verify the result by checking the units and ensuring that the calculated energy is reasonable for the given parameters. This step ensures that no errors were made during the calculation process.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field in a Solenoid

A solenoid is a coil of wire that generates a magnetic field when an electric current passes through it. The strength of the magnetic field (B) inside a long solenoid is given by the formula B = μ₀ * n * I, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. In this case, the solenoid produces a uniform magnetic field of 5.0 T due to the 100 A current.
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Magnetic Energy Storage

The magnetic energy (U) stored in a magnetic field can be calculated using the formula U = (1/2) * L * I², where L is the inductance of the solenoid and I is the current. The inductance depends on the solenoid's geometry and the number of turns of wire. This energy is crucial for understanding how much energy is stored in the magnetic field when the solenoid is energized.
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Superconductivity

Superconductivity is a phenomenon where certain materials exhibit zero electrical resistance when cooled below a critical temperature. In the context of the solenoid, using superconducting wires allows for the flow of large currents (like 100 A) without energy loss due to resistance. This is essential for maintaining the strong magnetic field required for MRI applications.
Related Practice
Textbook Question

A 2.0 mH inductor is connected in parallel with a variable capacitor. The capacitor can be varied from 100 pF to 200 pF. What is the range of oscillation frequencies for this circuit?

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Textbook Question

BIO An MRI machine needs to detect signals that oscillate at very high frequencies. It does so with an LC circuit containing a 15 mH coil. To what value should the capacitance be set to detect a 450 MHz signal?

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Textbook Question

Electricity is distributed from electrical substations to neighborhoods at 15,000 V. This is a 60 Hz oscillating (AC) voltage. Neighborhood transformers, seen on utility poles, step this voltage down to the 120 V that is delivered to your house. No energy is lost in an ideal transformer, so the output power Pout from the secondary coil equals the input power Pin to the primary coil. Suppose a neighborhood transformer delivers 250 A at 120 V. What is the current in the 15,000 V line from the substation?

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Textbook Question

How much energy is stored in a 3.0-cm-diameter, 12-cm-long solenoid that has 200 turns of wire and carries a current of 0.80 A?

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Textbook Question

What is the potential difference across a 10 mH inductor if the current through the inductor drops from 150 mA to 50 mA in 10 μs? What is the direction of this potential difference? That is, does the potential increase or decrease along the direction of the current?

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Textbook Question

The switch in FIGURE EX30.32 has been in position 1 for a long time. It is changed to position 2 at t = 0 s. What is the first time at which the current is maximum?

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