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 69a

30장, 문제 69a

A 50 cm solenoid with 1000 turns has an inductance of 20 mH. What is the magnetic field strength inside the inductor when the current is 75 mA?

검증된 단계별 안내

Step 1: Recall the formula for the inductance of a solenoid:

L = μ₀μr (N²A)/l

, where

L

is the inductance,

μ₀

is the permeability of free space,

μr

is the relative permeability,

N

is the number of turns,

A

is the cross-sectional area, and

l

is the length of the solenoid. However, for this problem, we are directly given the inductance and need to calculate the magnetic field strength.

Step 2: Use the formula for the magnetic field strength inside a solenoid:

B = μ₀μr (N/l)I

, where

B

is the magnetic field strength,

N

is the number of turns,

l

is the length of the solenoid, and

I

is the current.

Step 3: Convert the given values into SI units. The length of the solenoid is

l = 50 \, \(\text{cm}\) = 0.50 \, \(\text{m}\)

, the current is

I = 75 \, \(\text{mA}\) = 0.075 \, \(\text{A}\)

, and the number of turns is

N = 1000

. The permeability of free space is a constant:

μ₀ = 4π × 10^{-7} \, \(\text{T·m/A}\)

. Assume the relative permeability

μr

is 1 (for air or vacuum).

Step 4: Substitute the values into the formula for the magnetic field strength:

B = (4π × 10^{-7})(1000/0.50)(0.075)

. Simplify the expression step by step to calculate the magnetic field strength.

Step 5: Ensure the units are consistent and verify the calculation. The result will give the magnetic field strength

B

in teslas (T).

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주요 개념

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

Inductance

Inductance is a property of an electrical component, typically a coil or solenoid, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is defined as the ratio of the induced electromotive force (EMF) to the rate of change of current. In this case, the solenoid's inductance of 20 mH indicates its capacity to generate a magnetic field in response to the current.

Magnetic Field Strength

Magnetic field strength, often denoted as H, is a measure of the intensity of the magnetic field produced by an electric current. For a solenoid, it can be calculated using the formula H = nI, where n is the number of turns per unit length and I is the current. In this scenario, the magnetic field strength inside the solenoid can be determined by substituting the given values of current and the solenoid's dimensions.

Solenoid

A solenoid is a cylindrical coil of wire that generates a magnetic field when an electric current passes through it. The strength of the magnetic field inside a long solenoid is uniform and can be calculated using the formula B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. Understanding the properties of solenoids is essential for analyzing their behavior in electromagnetic applications.

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