Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 29: Electromagnetic Induction

문제 39

29장, 문제 39

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 × 10^-6 V/m. Calculate di/dt.

검증된 단계별 안내

Start by understanding the concept of electromagnetic induction in solenoids. When the current in a solenoid changes, it induces an electric field around it. This is described by Faraday's law of induction.

Use the formula for the induced electric field inside a solenoid:

E = \frac{r}{R} \frac{d}{i} d t

, where

E

is the induced electric field,

r

is the distance from the axis,

R

is the radius of the solenoid, and

\frac{d}{i} d t

is the rate of change of current.

Substitute the given values into the formula:

8.00 \times 10^{- 6}

for

E

3.50

cm for

r

, and

1.10

cm for

R

. Convert these distances to meters.

Rearrange the formula to solve for

\frac{d}{i} d t

\frac{d}{i} d t

= ERr.

Calculate the value of

\frac{d}{i} d t

using the rearranged formula and the substituted values.

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주요 개념

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

Faraday's Law of Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. In the context of a solenoid, the changing current alters the magnetic field inside, inducing an electric field around the solenoid. This principle is crucial for understanding how the rate of change of current (di/dt) affects the induced electric field.

Magnetic Field Inside a Solenoid

The magnetic field inside a long, thin solenoid is uniform and given by B = μ₀nI, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. This concept helps in understanding how the solenoid's properties and current contribute to the magnetic field, which is essential for calculating the induced electric field.

Induced Electric Field

The induced electric field around a solenoid is related to the rate of change of the magnetic field inside it. According to Faraday's Law, the magnitude of the induced electric field at a distance from the solenoid's axis can be calculated using the formula E = (r/2) * (dB/dt), where r is the radial distance from the axis. This concept is key to determining the relationship between di/dt and the given electric field.