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.

검증된 단계별 안내

Step 1: Calculate the area of the coil. The coil has a diameter of 5.0 cm, so the radius is 2.5 cm or 0.025 m. The area of the coil is given by the formula A = πr². Substitute the radius into the formula to find the area.

Step 2: Determine the magnetic flux through the coil. Magnetic flux (Φ) is given by Φ = N * B * A, where N is the number of turns, B is the magnetic field, and A is the area of the coil. Substitute the values for N, B, and A into the formula. Note that B is a function of time, B = 0.020t + 0.010t².

Step 3: Find the rate of change of magnetic flux (dΦ/dt). Differentiate the expression for Φ with respect to time t. Use the product rule if necessary, as B is time-dependent.

Step 4: Calculate the induced electromotive force (EMF) using Faraday's law of induction. Faraday's law states that EMF = -dΦ/dt. Substitute the result from Step 3 into this formula.

Step 5: Determine the current (I) in the coil using Ohm's law. Ohm's law states that I = EMF / R, where R is the resistance of the coil. Substitute the values for EMF and R to find the current at t = 5 s and t = 10 s.

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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 proportional to the rate of change of the magnetic flux, which can be calculated using the formula EMF = -dΦ/dt, where Φ is the magnetic flux. This principle is fundamental for understanding how coils generate current when exposed to changing magnetic fields.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. This law is crucial for calculating the current flowing through the coil once the induced EMF is known. By rearranging the formula, we can find the current as I = V/R, allowing us to determine how the resistance of the coil affects the current generated by the induced EMF.

Magnetic Flux

Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which it passes, taking into account the angle (θ) between the field lines and the normal to the surface: Φ = B·A·cos(θ). In this problem, the magnetic field varies with time, which affects the flux through the coil and is essential for calculating the induced EMF and resulting current at specific times.