Problem 34 - College Physics

Question

An inductor has inductance of 0.260 \(\mathrm{H}\) and carries a current that is decreasing at a uniform rate of 18.0 \(\mathrm{mA} / \mathrm{s} .\) Find the self-induced emf in this inductor.

Step-by-Step Solution

Answer

The self-induced emf is \( -0.00468 \ \text{V} \).

Step 1: Identify the formula for self-induced emf

The self-induced electromotive force (emf) in an inductor is given by the formula: \( \varepsilon = -L \frac{dI}{dt} \), where \( \varepsilon \) is the emf, \( L \) is the inductance, and \( \frac{dI}{dt} \) is the rate of change of current.

Step 2: Substitute the given values into the formula

Substitute the given values into the formula. The inductance \( L \) is \( 0.260 \ \text{H} \) and the rate of change of current \( \frac{dI}{dt} \) is \( 18.0 \ \text{mA/s} \). First, convert \( 18.0 \ \text{mA/s} \) to \( \text{A/s} \) by dividing by 1000: \( 18.0 \ \text{mA/s} = 0.018 \ \text{A/s} \). Substitute these values into the formula \( \varepsilon = -0.260 \times 0.018 \).

Step 3: Calculate the self-induced emf

Perform the multiplication to find the self-induced emf: \( \varepsilon = -0.260 \times 0.018 = -0.00468 \). The negative sign indicates that the emf opposes the change in current, according to Lenz's Law.

Key Concepts

Inductance

Inductance is a fundamental property of electrical circuits, particularly in coils or inductors. It describes the ability of a component to oppose changes in current. When we talk about inductance, we often refer to the symbol \( L \), which represents the inductance measured in Henrys (\( H \)). Inductance comes into play when there's a change in the current flowing through a coil. Because inductors store energy in a magnetic field, any change in current will create a self-induced emf (electromotive force) that opposes that change. This is a fascinating self-regulating ability of inductors.

• Inductors resist changes in current, similar to how capacitors resist changes in voltage.
• The unit of inductance, the Henry, signifies the amount of induction required to induce one volt across a coil with a current change of one ampere per second.

For example, in the given exercise, an inductor has an inductance of 0.260 \( H \), and this value is used to calculate the self-induced emf when the current changes.

Rate of Change of Current

The rate of change of current is crucial in determining the self-induced emf in an inductor. It describes how fast the current increases or decreases over time and is mathematically represented as \( \frac{dI}{dt} \).In the exercise, this rate is given as 18.0 mA/s. Before using this value in calculations, it's often necessary to convert into standard units, which for this case would be amperes per second (A/s). Hence, 18.0 mA/s becomes 0.018 A/s.

• The rate of change of current directly affects the magnitude of the self-induced emf. The higher the rate, the greater the emf.
• This change can either be an increase or a decrease. In the given problem, it is a decrease, leading to a negative emf.

Understanding this concept helps us appreciate how variations in current, even if small, are significant when calculating the resulting effects in a circuit.

Lenz's Law

Lenz's Law is a critical principle when dealing with inductors and self-induced emf. It states that an induced emf will always work to oppose the change in current that created it. This natural law is why the formula for self-induced emf includes a negative sign.This means that:

• When the current through an inductor decreases, the induced emf will be positive to try to maintain the current.
• Conversely, if the current increases, the induced emf will be negative to resist the change.

In our exercise example, the current is decreasing, which produces a negative emf as seen from the calculation: \( \varepsilon = -0.00468 \). This indicates the inductor's response to counteract the current decline. Lenz's Law helps us predict these behaviors and is critical in designing and understanding circuits that make use of inductors.