A 1000-turn coil of wire 1.0 cm in diameter is in a magnetic field that increases from 0.10 T to 0.30 T in 10 ms. The axis of the coil is parallel to the field. What is the emf of the coil?

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Step 1: Calculate the area of the coil using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius of the coil. The diameter is given as 1.0 cm, so the radius is \( r = \frac{1.0 \text{ cm}}{2} = 0.5 \text{ cm} = 0.005 \text{ m} \). Substitute \( r \) into the formula to find \( A \).

Step 2: Determine the change in magnetic flux \( \Delta \Phi \) through the coil. Magnetic flux is given by \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area. The change in flux is \( \Delta \Phi = (B_{\text{final}} - B_{\text{initial}}) \cdot A \). Substitute \( B_{\text{initial}} = 0.10 \text{ T} \), \( B_{\text{final}} = 0.30 \text{ T} \), and the area \( A \) calculated in Step 1.

Step 3: Use Faraday's law of electromagnetic induction to calculate the induced emf. Faraday's law states \( \text{emf} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns in the coil, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time interval. Substitute \( N = 1000 \), \( \Delta \Phi \) from Step 2, and \( \Delta t = 10 \text{ ms} = 0.010 \text{ s} \).

Step 4: Simplify the expression for the emf by performing the necessary multiplications and divisions. Ensure the negative sign from Faraday's law is included, which indicates the direction of the induced emf according to Lenz's law.

Step 5: Interpret the result. The magnitude of the emf represents the rate of change of magnetic flux through the coil, and the negative sign indicates the direction of the induced current opposes the change in magnetic flux.

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주요 개념

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

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a coil induces an electromotive force (emf) in the coil. The induced emf is directly proportional to the rate of change of magnetic flux and the number of turns in the coil. Mathematically, it is expressed as emf = -N (ΔΦ/Δt), where N is the number of turns, ΔΦ is the change in magnetic flux, and Δt is the time interval.

Magnetic Flux

Magnetic flux (Φ) is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass, given by Φ = B · A · cos(θ), where θ is the angle between the magnetic field lines and the normal to the surface. In this case, since the coil's axis is parallel to the field, θ is 0 degrees.

Induced EMF Calculation

To calculate the induced emf in the coil, we first determine the change in magnetic flux due to the change in the magnetic field. The area of the coil can be calculated using its diameter, and the change in magnetic field is the difference between the final and initial values. By substituting these values into Faraday's Law, we can find the induced emf, which quantifies the voltage generated in the coil due to the changing magnetic field.