Question

Two coils have mutual inductance \(M = 3.25 \times 10^{-4}\) H. The current \(i_1\) in the first coil increases at a uniform rate of 830 A/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?

Short answer

(a) 0.26975 V; it is constant. (b) 0.26975 V; it is also constant.

Step-by-step solution

Understanding Mutual Inductance

Mutual inductance \(M\) is a measure of how much the magnetic field created by one coil induces electromotive force (emf) in another coil. Here, \(M = 3.25 \times 10^{-4}\) H.

Analyze Induced EMF Formula

The emf induced in a coil is given by the formula \( \text{emf} = M \times \frac{di}{dt} \), where \( \frac{di}{dt} \) is the rate of change of current. Here, it is uniform at 830 A/s.

Calculate the Magnitude of Induced EMF for Part (a)

Using the formula \( \text{emf} = M \times \frac{di}{dt} \), substitute \(M = 3.25 \times 10^{-4}\) H and \(\frac{di_1}{dt} = 830\) A/s: \[ \text{emf} = 3.25 \times 10^{-4} \times 830 = 0.26975 \text{ V} \] Thus, the magnitude of the induced emf in the second coil is 0.26975 V.

Determine if the Induced EMF is Constant

Since the rate of change of current \(\frac{di_1}{dt}\) is constant, the induced emf is also constant at 0.26975 V.

Calculate the Magnitude of Induced EMF for Part (b)

For part (b), if the current changing at the same rate is in the second coil, the induced emf in the first coil is calculated similarly: Use the same formula, where \(M = 3.25 \times 10^{-4}\) H and \(\frac{di_2}{dt} = 830\) A/s: \[ \text{emf} = 3.25 \times 10^{-4} \times 830 = 0.26975 \text{ V} \]Thus, the induced emf in the first coil is also 0.26975 V, which is constant.

Key concepts

Induced Electromotive Force (EMF)

Induced electromotive force (emf) is a fundamental concept in electromagnetism. It refers to the voltage that is generated in a coil due to a changing magnetic field. This phenomenon is based on Faraday's law of electromagnetic induction. The law states that the induced emf in a coil is directly proportional to the rate of change of the magnetic flux through the coil. In simple terms, any time there is a change in the magnetic environment of a coil, it will cause an induced voltage in the coil.

For two mutually inductive coils, the induced emf in one coil can be calculated using the formula:

• \( \text{emf} = M \times \frac{di}{dt} \)

Here, \(M\) is the mutual inductance between the coils, and \(\frac{di}{dt}\) is the rate of change of current in the other coil. If the current's rate of change is uniform, like in our exercise, the induced emf will be constant. This is important because it means the induced voltage does not fluctuate, making it predictable and easier to manage in electrical circuits.

Rate of Change of Current

The rate of change of current is an essential factor in determining the induced emf across a coil. This rate, denoted by \( \frac{di}{dt} \), represents how quickly the current in a circuit is increasing or decreasing. In our context, it’s the speed at which the current in one coil changes that affects the second coil via mutual inductance.

Consider the given rate of current change, which is 830 A/s for our exercise. This rate is uniform, meaning it doesn't change over time. When the rate of current change is constant as in our case, it simplifies the calculation of induced emf because you can directly multiply it with the mutual inductance \(M\) to find the emf. The induced voltage remains steady over time because of this uniformity, which is a common scenario in many practical applications, such as transformers.

Magnetic Field

A magnetic field is an invisible field that exerts force on substances that are sensitive to magnetism, like iron and other ferromagnetic materials. In the context of mutual inductance, the magnetic field created by a current-carrying coil is key to inducing an emf in another nearby coil.

This magnetic field is generated when an electric current passes through a wire or coil. The strength and orientation of this magnetic field change with the current. So, when the current in one coil changes, it alters the magnetic field around it, which in turn affects the neighboring coil. This change is the driving factor that induces an emf in the second coil according to Faraday’s Law of Induction.

In applications involving mutual inductance, both coils influence one another through their magnetic fields. Understanding this interaction is crucial when designing circuits that rely on these principles, such as transformers, which adjust voltage levels in power systems.