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 26

Chapter 30, Problem 26

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?

Verified step by step guidance

Step 1: Calculate the magnetic field inside the solenoid using the formula for the magnetic field of a solenoid:

B = μ

₀nI, where

μ

₀ is the permeability of free space (

4 π \times 10 ⁻ 7

T·m/A),

n

is the number of turns per unit length, and

I

is the current.

Step 2: Determine the number of turns per unit length,

n

, using the formula

n = \frac{N}{L}

, where

N

is the total number of turns (200) and

L

is the length of the solenoid (12 cm, converted to meters).

Step 3: Calculate the energy density of the magnetic field using the formula

u = \frac{1}{2} μ

₀B², where

B

is the magnetic field calculated in Step 1.

Step 4: Calculate the volume of the solenoid using the formula for the volume of a cylinder:

V = π r ² L

, where

r

is the radius of the solenoid (half of the diameter, converted to meters) and

L

is the length of the solenoid.

Step 5: Multiply the energy density

u

by the volume

V

to find the total energy stored in the solenoid:

E = u V

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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 generates a magnetic field when an electric current passes through it. The strength of this magnetic field (B) inside a long solenoid can be calculated using the formula B = μ₀ * (N/L) * I, where μ₀ is the permeability of free space, N is the number of turns, L is the length of the solenoid, and I is the current. Understanding this concept is crucial for determining the energy stored in the magnetic field.

Energy Stored in a Magnetic Field

The energy (U) stored in the magnetic field of a solenoid can be calculated using the formula U = (1/2) * L * I², where L is the inductance of the solenoid. The inductance depends on the solenoid's geometry and the number of turns. This concept is essential for solving the question as it directly relates to the energy stored due to the current flowing through the solenoid.

Inductance of a Solenoid

The inductance (L) of a solenoid is a measure of its ability to store magnetic energy and is given by the formula L = (μ₀ * N² * A) / L, where A is the cross-sectional area of the solenoid. The diameter and length of the solenoid are critical in calculating the area and thus the inductance. This concept is vital for determining how much energy is stored in the solenoid when a current flows through it.

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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 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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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 P_out from the secondary coil equals the input power P_in 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

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

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