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Chapter 10: Problem 1

Two coils have mutual inductance \(M=3 \times 10^{-4} \mathrm{H} .\) The current \(i_{1}\) in the first coil increases at a uniform rate of \(800 \mathrm{~A} / \mathrm{s}\) (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?

Short Answer

Expert verified
(a) 0.24 V, yes, it's constant. (b) 0.24 V, it's also constant.

Step by step solution

01

Understanding Mutual Inductance

The concept of mutual inductance is essential to solving this problem. It indicates how much voltage is induced in one coil due to a change in current in another coil. The formula for induced emf due to mutual inductance is given by \( \varepsilon = -M \frac{di}{dt} \).
02

Calculate Induced EMF in Second Coil

To find the induced emf in the second coil when the current in the first coil changes, we use the formula \( \varepsilon = -M \frac{di_1}{dt} \). Given that \( M = 3 \times 10^{-4} \) H and \( \frac{di_1}{dt} = 800 \) A/s, substitute these values into the formula to get \( \varepsilon = -3 \times 10^{-4} \times 800 = -0.24 \) V. The magnitude is therefore \( 0.24 \) V. Since \( \frac{di_1}{dt} \) is constant, the induced emf is also constant.
03

Calculate Induced EMF in First Coil

If the current change occurs in the second coil instead, we still use \( \varepsilon = -M \frac{di_2}{dt} \). Here, \( \frac{di_2}{dt} \) is the same since it's given as \( 800 \) A/s. Applying the same values, \( \varepsilon = -3 \times 10^{-4} \times 800 = -0.24 \) V, with a magnitude of \( 0.24 \) V. It remains constant for the same reason.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Induced Electromotive Force (EMF)
Induced electromotive force, often abbreviated as emf, is a fundamental concept in electromagnetism. When current changes within a coil, it influences nearby coils through inductance. This phenomenon occurs due to mutual inductance.

Mutual inductance describes the process where a change in current in one coil induces a voltage in another. The relationship between these coils is quantified by mutual inductance. The equation for induced emf through mutual inductance is given by:
  • \( \varepsilon = -M \frac{di}{dt} \)
The negative sign indicates that the induced emf opposes the change in current, adhering to Lenz's Law.

This principle is crucial in transformers and other electrical devices where energy transfer occurs between coils without direct electrical connections.
Rate of Change of Current
The rate of change of current measures how quickly current varies over time. This is symbolized as \( \frac{di}{dt} \), representing the derivative of current with respect to time.

In the context of mutual inductance and induced emf, the rate of current change directly influences the magnitude of the induced emf. A higher rate of change leads to a larger emf, assuming all other factors remain constant. This relationship is described by:
  • High \( \frac{di}{dt} \) means stronger induced emf.
  • Constant \( \frac{di}{dt} \) results in a constant emf.
Understanding this concept is vital for navigating circuits involving inductors, transformers, and similar components.

It helps quantify how systems respond to dynamic changes, crucial for designing stable and efficient electrical networks.
Coils in Electrical Circuits
Coils, or inductors, are pivotal components in electrical circuits. They store energy in magnetic fields when electric current passes through them. Coils exhibit properties of self-inductance and mutual inductance.

In systems with more than one coil, like transformers, mutual inductance becomes significant. Coils can affect each other through induced emf if they are in close proximity. This interaction forms the basis of many modern electrical systems such as:
  • Transformers: for stepping voltage levels up or down.
  • Inductive sensors: for measuring displacement, pressure, or flow.
Comprehending how coils function and interact in circuits provides a deeper understanding of electromagnetic phenomena. It allows for the creation and manipulation of devices that depend on magnetic fields.

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Most popular questions from this chapter

In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of \(4.4 \mathrm{~T}\). What is the magnetic-field energy in a \(10.0-\mathrm{cm}^{3}\) volume of space where \(B=4.80 \mathrm{~T}\) ?

A toroidal solenoid has 500 turns, cross-sectional area \(6 \mathrm{~cm}^{2}\), and mean radius \(4.00 \mathrm{~cm}\). (a) Calculate the coil's selfinductance. (b) If the current decreases uniformly from \(5.00 \mathrm{~A}\) to \(2.00 \mathrm{~A}\) in \(3.00 \mathrm{~ms}\), calculate the self-induced emf in the coil. (c) The current is directed from terminal \(a\) of the coil to terminal \(b\). Is the direction of the induced emf from \(a\) to \(b\) or from \(b\) to \(a\) ?

A toroidal solenoid has a mean radius \(r\) and a crosssectional area \(A\) and is wound uniformly with \(N_{1}\) turns. A second toroidal solenoid with \(N_{2}\) turns is wound uniformly around the first. The two coils are wound in the same direction. (a) Derive an expression for the inductance \(L_{1}\) when only the first coil is used and an expression for \(L_{2}\) when only the second coil is used. (b) Show that \(M^{2}=L_{1} L_{2}\)

An inductor with an inductance of \(2.50 \mathrm{H}\) and a resistance of \(8.00 \Omega\) is connected to the terminals of a battery with an emf of \(6.00 \mathrm{~V}\) and negligible internal resistance. Find (a) the initial rate of increase of current in the circuit; (b) the rate of increase of current at the instant when the current is \(0.500 \mathrm{~A} ;\) (c) the current \(0.250 \mathrm{~s}\) after the circuit is closed; (d) the final steady-state current.

An inductor \(L\) and resistor \(R\) are connected in a single loop. The current reduces to zero from its maximum value. At time \(t=t_{0}\), the energy converted to heat equal the energy stored in the inductor at the time. Find the value of \(t_{0}\).
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