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

(a) 0.27 V; constant. (b) 0.27 V; constant.

Step 1: Understand the Concept of Mutual Inductance

Mutual inductance occurs when a change in current in one coil induces an electromotive force (emf) in another nearby coil. The magnitude of the induced emf (\[ |\varepsilon| = M \frac{di_1}{dt} \]where \( M \) is the mutual inductance, and \( \frac{di_1}{dt} \) is the rate of change of current in the first coil. It will be the same for induced emf in the other coil if roles are reversed.

Step 2: Calculate Induced EMF in the Second Coil (Part a)

Given: \( M = 3.25 \times 10^{-4} \text{ H} \) and \( \frac{di_1}{dt} = 830 \text{ A/s} \). Use the mutual inductance formula:\[\varepsilon = M \frac{di_1}{dt} = (3.25 \times 10^{-4} \text{ H})(830 \text{ A/s})\]\[\varepsilon = 0.26975 \text{ V} \approx 0.27 \text{ V}\]Therefore, the induced emf in the second coil is approximately 0.27 V, and it is constant because the rate of change of current is uniform.

Step 3: Calculate Induced EMF in the First Coil (Part b)

If the situation is reversed, the equation for induced emf remains the same because it is symmetrical:\[\varepsilon = M \frac{di_2}{dt} = (3.25 \times 10^{-4} \text{ H})(830 \text{ A/s})\]\[\varepsilon = 0.27 \text{ V}\]The induced emf in the first coil is also approximately 0.27 V, constant.

Key Concepts

Electromotive Force (EMF)

Electromotive Force, commonly abbreviated as EMF, is a fundamental concept in electromagnetism. It refers to the voltage generated by a source when no current is flowing. When we talk about EMF in circuits, it represents the energy provided by the power source to each charge carrier, like electrons.
Think of EMF as the 'push' behind electrical current. In the context of mutual inductance between coils, a changing magnetic field in one coil creates an EMF in another nearby coil. This effect is central to transformers and inductors in electronics.
It’s important to understand that while EMF can be thought of as a type of voltage, it doesn't always result in a current unless the circuit is closed. Analogously, you can think of EMF like water pressure; it causes the flow (current), but only if there’s a channel (closed path). Mutual inductance showcases how one coil can induce an EMF in another due to magnetic coupling.

Rate of Change of Current

The rate of change of current is a critical factor affecting induced EMF. Simply put, it describes how quickly the current is altering over time in a given circuit. Measured in amperes per second (A/s), it is crucial in understanding electromagnetic processes.
If you picture current as water flowing through a hose, the rate of change is akin to how quickly you turn up a faucet. In mutual inductance, a rapid change in current in one coil can induce a significant EMF in a second coil nearby.
This concept emphasizes that not just the amount of current matters, but how swiftly this current changes. By increasing the current speed, the induced EMF will proportionally rise, following the fundamental relation:

• \( rac{di}{dt} \) - rate of change of current.
• More change in less time results in higher induced EMF.
• Rapid changes have a greater impact on induction than slow alterations.

This principle is central to understanding how transformers function efficiently in altering voltages.

Induced EMF Calculation

Calculating the induced electromotive force (EMF) involves understanding mutual inductance and the rate at which the current changes. The basic formula to calculate EMF in a coil due to a change in current in a nearby coil is given by:

• \( \varepsilon = M \frac{di}{dt} \)

Where \( M \) is the mutual inductance (measured in henries, H), and \( \frac{di}{dt} \) is the rate of change of the current (in A/s).
This straightforward formula captures the linear relationship: double the rate of change, double the EMF; double the mutual inductance, double the EMF. For a given mutual inductance, as the current changes more rapidly, the induced EMF increases proportionally.
In practice, this means selecting components with appropriate inductance values to control EMF levels when designing electric circuits. It’s pivotal to ensure precise control over current changes, as these directly influence the EMF induced in interconnected components, impacting circuit performance.