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

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 31

Chapter 30, Problem 31

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?

Verified step by step guidance

Start by recalling the formula for the resonant frequency of an LC circuit:

f = \(\frac{1}{2\pi \sqrt{L C}\)}

, where

f

is the resonant frequency,

L

is the inductance, and

C

is the capacitance.

Rearrange the formula to solve for the capacitance

C

C = \(\frac{1}{(2\pi f)^2 L}\)

Substitute the given values into the formula. The inductance

L

is 15 mH, which is

15 \(\times\) 10^{-3}

H, and the frequency

f

is 450 MHz, which is

450 \(\times\) 10^{6}

Hz.

Calculate the term

(2\(\pi\) f)^2

by first finding

2\(\pi\) f

and then squaring the result.

Finally, divide

1

by the product of

(2\(\pi\) f)^2

and

L

to find the capacitance

C

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Key Concepts

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

LC Circuit

An LC circuit, consisting of an inductor (L) and a capacitor (C), is a fundamental electrical circuit used to generate oscillating signals. The inductor stores energy in a magnetic field, while the capacitor stores energy in an electric field. The oscillation frequency of the circuit is determined by the values of L and C, following the formula f = 1/(2π√(LC)), where f is the frequency in hertz.

Resonant Frequency

The resonant frequency is the frequency at which an LC circuit naturally oscillates when not driven by an external force. At this frequency, the inductive and capacitive reactances are equal in magnitude but opposite in phase, resulting in maximum energy transfer. For an MRI machine, tuning the LC circuit to the resonant frequency of the desired signal is crucial for effective signal detection.

Capacitance Calculation

To determine the required capacitance for a specific resonant frequency in an LC circuit, the formula C = 1/(4π²f²L) is used. Here, C is the capacitance in farads, f is the frequency in hertz, and L is the inductance in henries. By substituting the known values of frequency and inductance into this formula, one can calculate the necessary capacitance to achieve resonance at the desired signal frequency.

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Guided course

08:02

Capacitors & Capacitance (Intro)

Related Practice

Textbook Question

At t = 0 s, the current in the circuit in FIGURE EX30.35 is I0. At what time in μs is the current (1/2)I₀?

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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 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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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

A 100-turn, 2.0-cm-diameter coil is at rest with its axis vertical. A uniform magnetic field 60° away from vertical increases from 0.50 T to 1.50 T in 0.60 s. What is the induced emf in the coil?

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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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