Problem 44 - University Physics with Modern Physics

Question

A coil has 400 turns and self-inductance 3.50 \(\mathrm{mH}\) . The current in the coil varies with time according to \(i=\) \((680 \mathrm{mA}) \cos (\pi t)(0.0250 \mathrm{s}) \cdot(\mathrm{a})\) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil?(c) At \(t=0.0180 \mathrm{s}\) , what is the magnitude of the induced emf?

Step-by-Step Solution

Answer

The maximum induced emf is approximately 29.936 V, maximum flux per turn is 5.95 μWb, and the emf at t=0.0180 s is about -19.90 V.

Step 1: Write the Induced EMF Formula

The emf (electromotive force) induced in a coil is given by Faraday's Law as \( \varepsilon = -L \frac{di}{dt} \), where \( L \) is the self-inductance of the coil and \( \frac{di}{dt} \) is the rate of change of current with respect to time. This equation will help find the maximum induced emf.

Step 2: Determine the Rate of Change of Current

Given that the current \( i = (680 \mathrm{mA}) \cos(\pi t/0.0250 \mathrm{s}) \), we need to find \( \frac{di}{dt} \). Differentiation gives: \( \frac{di}{dt} = \frac{-680 \mathrm{mA}}{0.0250 \mathrm{s}} \pi \sin(\pi t/0.0250 \mathrm{s}) \).

Step 3: Find Maximum Induced EMF

The term \( \sin(\pi t/0.0250 \mathrm{s}) \) achieves its maximum value of 1, which gives the maximum \( \frac{di}{dt} \). Substituting into the emf formula we get: \( \varepsilon_{\text{max}} = L \cdot \frac{680 \mathrm{mA}}{0.0250 \mathrm{s}} \cdot \pi \). With \( L = 3.50 \mathrm{mH} \), calculate \( \varepsilon_{\text{max}} \).

Step 4: Compute Maximum Average Flux per Turn

From Lenz's and Faraday's laws, the maximum flux \( \Phi \) can be expressed as \( L \cdot i_{\text{max}}/N \), where \( i_{\text{max}} = 680 \mathrm{mA} \) and \( N = 400 \) is the number of turns. Calculate \( \Phi_{\text{max}} \) using these values.

Step 5: Calculate Induced EMF at Specific Time

Use the previously determined \( \frac{di}{dt} \) formula. Substitute \( t = 0.0180 \mathrm{~s} \) into \( \frac{di}{dt} = \frac{-680 \mathrm{mA}}{0.0250 \mathrm{s}} \pi \sin(\pi t/0.0250 \mathrm{s}) \) and calculate \( \sin(\pi \times 0.0180/0.0250 ) \). Substitute this value into \( \varepsilon = -L \frac{di}{dt} \) to find the emf at \( t = 0.0180 \mathrm{s} \).

Key Concepts

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how electric currents can be induced by changing magnetic fields. It's a key concept in electromagnetism, crucial for understanding how devices like transformers and inductors work.

In most simple terms, Faraday's Law states that the induced electromotive force (emf) in a closed circuit is proportional to the rate of change of magnetic flux through the circuit. Mathematically, this is expressed as: \[ \varepsilon = - \frac{{d\Phi}}{{dt}} \] where \( \varepsilon \) is the induced emf and \( \frac{{d\Phi}}{{dt}} \) is the rate of change of magnetic flux.

• This equation reveals how a quick change in the magnetic environment can create electrical currents.
• The negative sign in the formula represents Lenz's Law, which states that the induced emf will generate a current that opposes the initial change in magnetic flux.

One common application involves coils of wire where Faraday's Law helps us calculate the induced emf when the magnetic field around the coil changes.

In our problem, we used Faraday's Law to find the induced emf by considering the change in current in a coil. The self-inductance of the coil helps in determining the induced emf along with how the current changes over time.

Self-Inductance

Self-inductance is a property of a coil or circuit whereby a change in current induces an emf in the same circuit. This phenomenon is rooted in Faraday's Law, and it becomes significant in scenarios involving coils and solenoids.

When you pass a current through a coil, it creates a magnetic field. If the current changes, the magnetic field also changes, inducing an emf in the coil itself. The self-inductance \( L \) of a coil is typically measured in henries (H) and describes how effectively a coil can induce emf from a change in its own current.

Mathematically, the induced emf \( \varepsilon \) due to self-inductance is given by: \[ \varepsilon = -L \frac{{di}}{{dt}} \] where \( L \) is the self-inductance, and \( \frac{{di}}{{dt}} \) represents the rate of change of current.

• The larger the self-inductance, the greater the emf for a given rate of change of current.
• This property allows inductors to resist changes in current, which can be useful in various electrical applications.

Self-inductance plays a crucial role in the exercise as it helps calculate the induced emf in the coil given the rate at which current changes. It highlights the inherent ability of the coil to respond to the current's variations.

Rate of Change of Current

The rate of change of current is a vital concept in electromagnetism that measures how quickly current levels are shifting in a circuit over time. This concept is essential for calculating induced emf in a coil, as per Faraday's Law and self-inductance principles.

Understanding and calculating the rate of change of current, \( \frac{{di}}{{dt}} \), requires differentiation when current as a function of time is given. This is because the electromagnetic effects depend on how rapidly the current changes, not just the amount of current.

For example, given a current function like \( i(t) = (680 \mathrm{mA}) \cos(\pi t/0.0250 \mathrm{s}) \), you differentiate it with respect to time \( t \) to find: \[ \frac{{di}}{{dt}} = \frac{-680 \mathrm{mA}}{0.0250 \mathrm{s}} \pi \sin(\pi t/0.0250 \mathrm{s}) \]

• This formula highlights the sinusoidal nature of current changes, which are common in AC circuits.
• A larger rate of change means a stronger induced emf, hence the importance of accurate differentiation.

This exercise used the rate of change of current to calculate the maximum emf induced in the coil and the emf at specific time points, showcasing how the dynamic nature of electrical currents leads to significant electromagnetic effects.