The current i through a 4.6 H inductor varies with time t as shown by the graph of Fig. 30-27, w

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The current i through a 4.6 H inductor varies with time t as...

The current $i$ through a $4.6 \mathrm{H}$ inductor varies with time $t$ as shown by the graph of Fig. $30-27$, where the vertical axis scale is set by $i_{s}=16 \mathrm{~A}$ and the horizontal axis scale is set by $t_{s}=6.0 \mathrm{~ms}$. The inductor has a resistance of $12 \Omega$. Find the magnitude of the induced emf $\mathscr{E}$ during time intervals (a) 0 to $2 \mathrm{~ms}$, (b) $2 \mathrm{~ms}$ to $5 \mathrm{~ms}$, and (c) $5 \mathrm{~ms}$ to $6 \mathrm{~ms}$. (Ignore the behavior at the ends of the intervals.)

Key Concepts

Self-Inductance

Self-inductance is a property of an inductor that quantifies its ability to oppose changes in the current passing through it. It is manifested by an induced electromotive force (emf) that is proportional to the rate at which the current changes. This concept is fundamental in circuit analysis involving inductors and is governed by the relation ? = -L (di/dt), where L is the inductance and di/dt is the time derivative of the current.

Faraday’s Law and Lenz’s Law

Faraday’s law of electromagnetic induction states that a changing magnetic flux through a circuit induces an emf in the circuit. Lenz’s law provides the direction of this induced emf, asserting that it opposes the change in current that created it. Together, these laws explain how inductors resist sudden changes in current, which is central to analyzing circuits with time-varying currents.

Rate of Change of Current

The rate of change of current (di/dt) is the key quantitative measure in finding the induced emf in an inductor. When current changes over time, its derivative, di/dt, characterizes how fast that change occurs. In practical problems, especially those involving graphical data, determining the slope of the current vs. time curve over given intervals allows for the computation of the induced emf without requiring detailed circuit equations.

Graphical Interpretation for Circuit Analysis

Graphical analysis involves interpreting a plot of current versus time to determine the slope (di/dt) in various intervals. This technique is crucial when the current changes in pieces or segments rather than following a single continuous function. It allows engineers and physicists to directly extract the rate of change information necessary for calculating induced emf as per Faraday’s law.

Internal Resistance of an Inductor

Real inductors typically have an internal resistance that affects the overall behavior of the circuit. This resistance causes a voltage drop that must be considered separately from the induced emf due solely to the changing magnetic field. While the induced emf is primarily determined by the inductance and the rate of change of current, internal resistance plays a role in the total voltage across the inductor and can influence transient responses in the circuit.

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