Problem 1 - University Physics with Modern Physics

Question

Two coils have mutual inductance \(M=3.25 \times 10^{-4} \mathrm{H}\) The current \(i_{1}\) in the first coil increases at a uniform rate of 830 \(\mathrm{A} / \mathrm{s} .\) (a) What is the magnitude of the induced emf in the second coil? Is it constant? (b) Suppose that the current described is in the second coil rather than the first. What is the magnitude of the induced emf in the first coil?

Step-by-Step Solution

Answer

The magnitude of the induced emf is 0.27 V, and it is constant for both cases.

Step 1: Understand the problem

In this exercise, we need to calculate the induced electromotive force (emf) in a coil when the current in an adjacent coil changes. The key concept is mutual inductance, which describes how the change in current in one coil induces an emf in another. The formula to calculate the induced emf due to mutual inductance is given by Faraday's law of induction: \( \varepsilon = -M \times \frac{di}{dt} \).

Step 2: Identify given values

We are given the mutual inductance, \( M = 3.25 \times 10^{-4} \text{ H} \), and the rate of change of current in the first coil as \( \frac{di_{1}}{dt} = 830 \text{ A/s} \). Using these values, we will find the induced emf in part (a).

Step 3: Calculate induced emf in the second coil (Part a)

Using the formula \( \varepsilon = -M \times \frac{di_{1}}{dt} \), substitute \( M = 3.25 \times 10^{-4} \text{ H} \) and \( \frac{di_{1}}{dt} = 830 \text{ A/s} \):\[ \varepsilon = -(3.25 \times 10^{-4}) \times 830 \]Calculate to find \( \varepsilon = -0.26975 \text{ V} \). The minus sign indicates the direction of the induced emf opposes the change in current. The magnitude of the emf is 0.27 V, and it is constant because the rate of change of current is constant.

Step 4: Exchange the coils and reassess (Part b)

Part b asks us what happens if the current is in the second coil instead of the first. The mutual inductance is the same, \( M = 3.25 \times 10^{-4} \text{ H} \), and the current changes at the same rate \( \frac{di_{2}}{dt} = 830 \text{ A/s} \). Using \( \varepsilon = -M \times \frac{di_{2}}{dt} \),\[ \varepsilon = -(3.25 \times 10^{-4}) \times 830 \]Calculate to get \( \varepsilon = -0.26975 \text{ V} \). As before, the magnitude is 0.27 V and it remains constant.

Key Concepts

Faraday's law of induction

Faraday's law of induction is a key principle for understanding how electromagnetic fields interact with electric circuits. This law is fundamental in the study of electromagnetism. It states that a change in the magnetic field within a closed loop induces an electromotive force (emf) in the loop.
A simple way to picture this is to think about a coil of wire placed within a changing magnetic field. According to Faraday's law, this changing magnetic field will induce a voltage across the ends of the coil.

• The induced emf is directly related to the rate of change of the magnetic field. This means that the faster the field changes, the greater the induced emf.
• Faraday's law is mathematically expressed as \( \varepsilon = - \frac{d \Phi_B}{dt} \), where \( \varepsilon \) is the induced emf and \( \Phi_B \) is the magnetic flux.
• The negative sign, known as Lenz's Law, indicates that the induced emf creates a current whose magnetic field opposes the change in the original magnetic field.

Understanding this concept helps us explain many electrical technologies, from electric generators to transformers, which rely on principles of induced emf.

Induced Electromotive Force

The induced electromotive force, often abbreviated as emf, is an essential concept in electromagnetism, describing the voltage generated by a change in magnetic flux. This change typically occurs when a coil, such as one in an electrical circuit, experiences a varying magnetic field.

The induced emf is central to the operation of various electrical devices. For example, in an electric generator, mechanical energy is converted into electrical energy through the induction of emf.

• The formula to calculate induced emf, when considering mutual inductance, is \( \varepsilon = -M \times \frac{di}{dt} \). Here, \( M \) represents the mutual inductance between two coils, and \( \frac{di}{dt} \) is the rate of change of current through one of the coils.
• An induced emf can drive current when the circuit is complete, and its value depends on several factors, including the mutual inductance and the rate of current change.
• While often discussed in terms of coils and circuits, the concept of induced emf also applies to larger systems, highlighting the universality of electromagnetic principles.

Developing a solid grasp of induced emf is crucial for understanding how various electromagnetic systems and devices function.

Rate of Change of Current

In electrical systems involving mutual inductance, the rate of change of current plays a critical role. The rate at which current changes, represented as \( \frac{di}{dt} \), determines how quickly the magnetic field in a coil changes, which in turn affects the induced emf.

The concept is straightforward:

• A higher rate of change in current leads to a stronger induced emf. For instance, if the current through a coil increases rapidly, the resulting emf will be more significant.
• On the flip side, a slower rate of change in current results in a weaker induced emf. This relationship is why devices designed to manipulate or regulate current often pay close attention to how quickly current levels rise and fall.
• The rate of change is crucial in applications where timing and power control are necessary, such as in transformers and inductors, which rely on varying current to control energy transfer and storage efficiently.

By appreciating the rate of change of current, we gain insights into the dynamics of electromagnetic interactions and how they can be harnessed to serve our technological needs.