Problem 33 - College Physics

Question

At the instant when the current in an inductor is increasing at a rate of 0.0640 \(\mathrm{A} / \mathrm{s}\) , the magnitude of the self-induced emf is 0.0160 \(\mathrm{V} .\) What is the inductance of the inductor?

Step-by-Step Solution

Answer

The inductance is 0.25 henrys.

Step 1: Identify the Given Values

We are given that the rate of change of current, \( \frac{di}{dt} \), is 0.0640 A/s, and the induced emf, \( \varepsilon \), is 0.0160 V.

Step 2: Use the Formula for Induced EMF

We know from Faraday's law of electromagnetic induction that the magnitude of the self-induced emf (\( \varepsilon \)) in an inductor is given by the formula \( \varepsilon = L \cdot \frac{di}{dt} \), where \( L \) is the inductance.

Step 3: Rearrange the Equation to Solve for Inductance

Rearrange the formula \( \varepsilon = L \cdot \frac{di}{dt} \) to solve for \( L \):\[L = \frac{\varepsilon}{\frac{di}{dt}}\]

Step 4: Substitute the Given Values

Substitute the known values into the equation: \( L = \frac{0.0160 \, \text{V}}{0.0640 \, \text{A/s}} \).

Step 5: Calculate the Inductance

Perform the division: \( L = \frac{0.0160}{0.0640} = 0.25 \).

Step 6: Conclusion

The inductance of the inductor is 0.25 henrys.

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle in electromagnetism. It describes how electric currents are induced in a conductor due to changing magnetic fields. Imagine a magnetic field passing through a loop of wire. When the strength of this magnetic field changes, it creates an electromotive force (EMF) in the wire. This is a brilliant concept because it lays the groundwork for how we understand generators and transformers.

In this context, the law can be described mathematically as:

• The induced EMF is proportional to the rate of change of the magnetic flux.
• This relationship is captured by the equation \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced EMF and \( \Phi \) is the magnetic flux.
• The negative sign shows that the induced current will oppose the change in magnetic flux, as per Lenz's Law.

In our scenario of an increasing current, Faraday's Law specifically helps relate this changing magnetic field to the observed self-induced EMF.

Self-induced EMF

Self-induced EMF occurs when the changing current in a coil affects its own magnetic field, inducing an EMF that opposes the change of the current. It is one of the key aspects of an inductor's function. When you have a coil and current flowing through it, the current generates a magnetic field.

• As the current changes, so does the magnetic field, inducing a voltage across the coil.
• This self-induced EMF follows the same fundamental principle as Faraday's Law but happens within the conductor rather than across multiple wires or conductors.
• The formula \( \varepsilon = L \cdot \frac{di}{dt} \) shows this relationship, where \( L \) is the inductance and \( \frac{di}{dt} \) is the rate of change of current.

In practical applications, this self-induction is crucial in devices such as electrical transformers and chokes, as it helps manage and stabilize current flows.

Rate of change of current

The rate of change of current, expressed as \( \frac{di}{dt} \), is essential in understanding how quickly the current flowing through an inductor is increasing or decreasing. This rate directly influences the self-induced EMF in a conductor. Here’s how it works:

• A rapid increase in current can lead to a stronger self-induced EMF, which may oppose this change more significantly.
• The magnitude of self-induced EMF depends on both the rate of change of the current and the inductance of the coil.
• This means devices like inductors need to be carefully designed to handle specific current changes effectively, preventing unwanted spikes that could harm electrical circuits.

Understanding \( \frac{di}{dt} \) is crucial for anyone working with electric circuits, helping to predict how changes in current will affect other components.