Inductance and Induced EMF Calculation

This problem focuses on understanding the relationship between inductance, the rate of change of current, and the induced electromotive force (EMF) in an inductor. This is a core concept in electromagnetism, specifically governed by Faraday's Law of Induction and Lenz's Law.

Understanding Key Concepts

• Inductance (L): Inductance is a property of an electrical conductor or circuit that measures how much an electric current flowing through it creates a magnetic flux. It is measured in Henries (H).
• Induced EMF (ε): When the magnetic flux through a coil changes, an electromotive force is induced in the coil. For an inductor, the induced EMF is directly proportional to the rate of change of current flowing through it. The formula for the induced EMF in an inductor is:

\(\varepsilon = -L \frac{dI}{dt}\)

• Here, \(\varepsilon\) is the induced EMF, \(L\) is the inductance, and \(\frac{dI}{dt}\) is the rate of change of current. The negative sign represents Lenz's Law, indicating that the induced EMF opposes the change in current that caused it. However, for calculations involving the magnitude of the induced EMF, we often use the absolute value:

\(|\varepsilon| = L \left| \frac{\Delta I}{\Delta t} \right|\)

• In this problem, the current changes linearly, so \(\frac{\Delta I}{\Delta t}\) is the average rate of change of current.

Given Information

Let's list the known values provided in the problem statement:

• Inductance of the coil (\(L\)) = 5 H
• Initial current (\(I_1\)) = 10 A
• Final current (\(I_2\)) = 2 A
• Magnitude of induced EMF (\(|\varepsilon|\)) = 40 V

Calculating the Change in Current

First, we need to determine the total change in current (\(\Delta I\)) that occurs over the time period \(t\):

\(\Delta I = I_{\text{final}} - I_{\text{initial}}\)

\(\Delta I = 2 \text{ A} - 10 \text{ A}\)

\(\Delta I = -8 \text{ A}\)

Since the problem provides the magnitude of the induced EMF, we will use the magnitude of the change in current, which is \(|\Delta I| = |-8 \text{ A}| = 8 \text{ A}\).

Applying the Induced EMF Formula

Now, we can use the formula for the magnitude of the induced EMF in an inductor:

\(|\varepsilon| = L \frac{|\Delta I|}{t}\)

Substitute the given values into the formula:

\(40 \text{ V} = 5 \text{ H} \times \frac{8 \text{ A}}{t}\)

Solving for Time (t)

Let's simplify and solve the equation for \(t\):

\(40 = \frac{5 \times 8}{t}\)

\(40 = \frac{40}{t}\)

To find the value of \(t\), we can multiply both sides of the equation by \(t\) and then divide by 40:

\(40t = 40\)

\(t = \frac{40}{40}\)

\(t = 1 \text{ s}\)

Final Answer Summary

The time \(t\) taken for the current to change linearly from 10 A to 2 A, given the inductance and induced EMF, is 1 second.

This detailed step-by-step calculation demonstrates how to apply the principles of electromagnetic induction to solve for an unknown variable like time, given other parameters related to the coil and current change.