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 20

30장, 문제 20

The magnetic field inside a 5.0-cm-diameter solenoid is 2.0 T and decreasing at 4.0 T/s. What is the electric field strength inside the solenoid at a point (a) on the axis and (b) 2.0 cm from the axis?

검증된 단계별 안내

Understand the problem: The solenoid has a magnetic field that is decreasing over time, which induces an electric field according to Faraday's Law of Induction. We need to calculate the electric field strength at two points: (a) on the axis of the solenoid and (b) at a distance of 2.0 cm from the axis.

Apply Faraday's Law of Induction: The induced electric field is related to the rate of change of the magnetic flux. The magnetic flux through a loop of radius \( r \) is \( \Phi_B = B \cdot A = B \cdot \pi r^2 \), where \( B \) is the magnetic field and \( A \) is the area of the loop.

Differentiate the magnetic flux with respect to time: The rate of change of flux is \( \frac{d\Phi_B}{dt} = \pi r^2 \cdot \frac{dB}{dt} \), where \( \frac{dB}{dt} \) is the rate of change of the magnetic field (given as \( -4.0 \ \text{T/s} \)).

Relate the induced electric field to the rate of change of flux: The magnitude of the induced electric field at a distance \( r \) from the axis is given by \( E = \frac{1}{2\pi r} \cdot \left| \frac{d\Phi_B}{dt} \right| = \frac{1}{2} \cdot r \cdot \left| \frac{dB}{dt} \right| \). Substitute \( r = 0 \ \text{m} \) for part (a) and \( r = 0.02 \ \text{m} \) for part (b).

Substitute the known values: For part (a), since \( r = 0 \), the electric field on the axis is zero because the loop area is zero. For part (b), substitute \( r = 0.02 \ \text{m} \) and \( \frac{dB}{dt} = -4.0 \ \text{T/s} \) into the formula \( E = \frac{1}{2} \cdot r \cdot \left| \frac{dB}{dt} \right| \) to calculate the electric field strength.

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주요 개념

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

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. In this context, the magnetic field inside the solenoid is given as 2.0 T (Tesla), which indicates the strength of the magnetic field. The behavior of the magnetic field is crucial for understanding how it interacts with electric fields and charges.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop. The rate of change of the magnetic field is essential for calculating the induced electric field. In this case, the magnetic field is decreasing at a rate of 4.0 T/s, which will generate an electric field inside the solenoid.

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. The strength of the electric field induced by the changing magnetic field can be calculated using the relationship defined by Faraday's Law. The electric field strength varies with distance from the axis of the solenoid, which is important for determining the values at different points inside the solenoid.

관련 실천

교과서 질문

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.

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교과서 질문

FIGURE EX30.19 shows the current as a function of time through a 20-cm-long, 4.0-cm-diameter solenoid with 400 turns. Draw a graph of the induced electric field strength as a function of time at a point 1.0 cm from the axis of the solenoid.

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교과서 질문

A 12-cm-diameter, 1.0-m-long solenoid is wound with 2000 turns of superconducting wire. When the magnet is turned on, the current increases from 0 to I_max in 2.5 s. At t = 1.0 s, the induced electric field midway between the axis and the windings is 7.5×10⁻³ V/m. What is the solenoid's steady magnetic field strength?

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교과서 질문

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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교과서 질문

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Evaluate I at t = 5 s and t = 10 s.