Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

문제 26

30장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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