Question

A coil has 400 turns and self-inductance 7.50 mH. The current in the coil varies with time according to \(i = (680 \, \mathrm{mA}) \mathrm{cos} (\pi{t}/0.0250 \, \mathrm{s})\). (a) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil? (c) At \(t = 0.0180\) s, what is the magnitude of the induced emf?

Short answer

(a) Maximum emf is 0.640 V. (b) Maximum average flux per turn is 1.28 x 10^-5 Wb. (c) Induced emf at 0.0180 s is 0.384 V.

Step-by-step solution

Understanding the problem

We need to find the maximum emf induced in the coil, the maximum average magnetic flux through each turn, and the magnitude of the induced emf at a specific time.

Understanding induced emf formula

The induced emf in a coil is given by Faraday's law as \( \epsilon = -L \frac{di}{dt} \), where \( L \) is the self-inductance and \( \frac{di}{dt} \) is the rate of change of current.

Calculate maximum induced emf

First, we express the current as \( i(t) = 680 \times 10^{-3} \cos\left(\frac{\pi t}{0.0250}\right) \). To find \( \frac{di}{dt} \), differentiate \( i(t) \), giving \( \frac{di}{dt} = -680 \times 10^{-3} \times \frac{\pi}{0.0250} \sin\left(\frac{\pi t}{0.0250}\right) \). The maximum change occurs when \( \sin \) is maximized at 1.

Calculate maximum di/dt

Plug in \( \sin = 1 \) to get the maximum rate of change of current: \( \left(\frac{di}{dt}\right)_{max} = 680 \times 10^{-3} \times \frac{\pi}{0.0250} \).

Calculate maximum emf

Using \( \epsilon = -L \frac{di}{dt} \), substitute in \( L = 7.50 \times 10^{-3} \) H and \( \left(\frac{di}{dt}\right)_{max} \): \( \epsilon_{max} = -7.50 \times 10^{-3} \times 680 \times 10^{-3} \times \frac{\pi}{0.0250} = \pi \times 0.204 \approx 0.640 \text{ V}\,\).

Determine maximum average flux

Use \( N \Phi = Li_{max} \). Substituting \( i_{max} = 0.680 \) A and \( L = 7.50 \times 10^{-3} \) H, we find \( \, N \Phi = 7.50 \times 10^{-3} \times 0.680 = 5.1 \times 10^{-3} \), so \( \Phi_{max} = \frac{5.1 \times 10^{-3}}{400} = 1.28 \times 10^{-5} \, \text{Wb}\).

Calculate induced emf at specific time

Find \( \sin(\frac{\pi \times 0.0180}{0.0250}) \) and substitute in \( \frac{di}{dt} \): \( \left(\frac{di}{dt}\right)_{t=0.0180} = -680 \times 10^{-3} \times \frac{\pi}{0.0250} \times \sin(2.261) \approx -25.51 \sin(2.261) \), and then calculate \( \epsilon = -L \frac{di}{dt} \approx 7.50 \times 10^{-3} \times 14.03 \approx 0.384 \, \text{V} \).

Key concepts

Self-inductance

Self-inductance is a key concept in electromagnetic induction. It refers to the ability of a coil or circuit to oppose changes in current flowing through it. Picture self-inductance as the coil's inertia against the shifting flow of electrons.

When current flows through a coil, it generates a magnetic field. Self-inductance is the reason this changing magnetic field induces an electromotive force (emf) within the coil itself. This is the coil's way of resisting changes in current. The self-inductance (L) of a coil is measured in henrys (H). In the problem, a coil with self-inductance of 7.50 mH demonstrates this phenomenon.

The induced emf in a coil is calculated using the formula: \( \epsilon = -L \frac{di}{dt} \). This equation shows the relationship between self-inductance L and the rate of change of current \( \frac{di}{dt} \). The negative sign arises from Lenz's law, indicating that the induced emf opposes the change in current.

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that governs how electric currents and magnetic fields interact. According to Faraday's Law, the induced emf in any closed circuit is equal to the negative of the rate of change of magnetic flux through the circuit.

This law can be expressed as: \( \epsilon = -\frac{d\Phi}{dt} \), where \( \Phi \) represents magnetic flux. In simple terms, whenever there is a change in magnetic flux, it excites an electric field which is experienced as emf in the circuit.

Consider the coil from the example - as the current changes, it alters the magnetic field around each turn, inducing an emf. This induced emf can be calculated using the provided forms or rearranged in terms of self-inductance, \( \epsilon = -L \frac{di}{dt} \). Thus, Faraday's Law connects variations in magnetic flux to generating electric fields.

Magnetic Flux

Magnetic flux is the measure of the amount of magnetic field passing through a given area, such as a loop of wire. Think of magnetic flux as the number of magnetic field lines penetrating through a surface.

In our exercise, magnetic flux \( \Phi \) is related to self-inductance and current. The formula is \( N\Phi = Li \), where N is the number of turns in the coil, \Phi is the magnetic flux through each turn, L is the self-inductance, and i is the current.

Therefore, if you know the self-inductance and the maximum current through the coil, you can use this formula to determine the maximum average magnetic flux through each turn. In practice, to find the average flux, it's computed as \( \Phi = \frac{Li}{N} \). For example, with a maximum current of 680 mA, the maximum magnetic flux through a single turn can be determined.

Rate of Change of Current

The rate of change of current is a pivotal factor in electromagnetic induction processes. It describes how quickly the current in a circuit or coil changes over time. This is crucial because the faster the current changes, the larger the induced emf.

In our problem, the current as a function of time is given by \( i = 680 \times 10^{-3} \cos\left(\frac{\pi t}{0.0250}\right) \). The rate of change of this current is found by differentiating it with respect to time: \( \frac{di}{dt} = -680 \times 10^{-3} \times \frac{\pi}{0.0250} \sin\left(\frac{\pi t}{0.0250}\right) \).

The rate reaches its maximum when the sine component is equal to 1. This means the magnitude of \( \frac{di}{dt} \), and thus the induced emf, is at its peak. Understanding how the rate of change of current affects emf helps better grasp the dynamics of circuits experiencing changing currents.