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Chapter 7: Problem 53

Two coils are wrapped around a cylindrical form in such a way that the same flux passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating the flux.) The "primary" coil has \(N_{1}\) turns and the secondary has \(N_{2}\) (Fig. 7.54). If the current \(I\) in the primary is changing. show that the emf in the secondary is given by $$\frac{\mathcal{E}_{2}}{\mathcal{E}_{1}}=\frac{N_{2}}{N_{1}}$$ where \(\mathcal{E}_{1}\) is the (back) emf of the primary. [This is a primitive transformer-a device for raising or lowering the emf of an alternating current source. By choosing the appropriate number of turns, any desired secondary emf can be obtained. If you think this violates the conservation of energy, check out Prob. 7.54.]

Short Answer

Expert verified
\(\frac{\mathcal{E}_2}{\mathcal{E}_1} = \frac{N_2}{N_1}\) by comparing induced emfs using Faraday's law.

Step by step solution

01

Understanding the Problem

We have two coils around a cylindrical form, with the same magnetic flux passing through both. The primary coil has \(N_1\) turns, and the secondary coil has \(N_2\) turns. The current \(I\) in the primary coil is changing, inducing an electromotive force (emf) in both coils. We aim to find the relationship between the emf in the secondary, \(\mathcal{E}_2\), and that in the primary, \(\mathcal{E}_1\).
02

Applying Faraday's Law of Induction

Faraday's Law states that the induced emf \(\mathcal{E}\) in a coil is equal to the negative rate of change of the magnetic flux times the number of turns: \(\mathcal{E} = -N\frac{d\Phi}{dt}\). For the primary coil, the emf is \(\mathcal{E}_1 = -N_1 \frac{d\Phi}{dt}\). For the secondary coil, the emf is \(\mathcal{E}_2 = -N_2 \frac{d\Phi}{dt}\).
03

Relating the Induced EMFs

Since both coils experience the same change in flux \(\frac{d\Phi}{dt}\), we can set up a ratio of the emfs: \(\frac{\mathcal{E}_2}{\mathcal{E}_1} = \frac{-N_2 \frac{d\Phi}{dt}}{-N_1 \frac{d\Phi}{dt}}\). The rate of change of flux cancels out, simplifying the expression to \(\frac{\mathcal{E}_2}{\mathcal{E}_1} = \frac{N_2}{N_1}\).
04

Interpreting the Result

This means that the ratio of the induced emf in the secondary to that in the primary coil is directly proportional to the ratio of their respective turns. This is a fundamental property of transformers and shows how changing the number of turns can increase or decrease voltage.

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Key Concepts

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

Faraday's law of induction
Faraday's law of induction is a fundamental principle that describes how a change in magnetic flux can induce an electromotive force (emf) in a coil of wire.

In simpler terms, if you have a coil and you change the magnetic environment around it, an electric potential (or voltage) gets generated.

This is written mathematically as \(\mathcal{E} = -N \frac{d\Phi}{dt}\), where:
  • \(\mathcal{E}\) is the induced emf,
  • \(N\) is the number of turns in the coil,
  • \(\Phi\) represents the magnetic flux, and
  • \(\frac{d\Phi}{dt}\) is the rate of change of magnetic flux over time.
The negative sign in the equation represents Lenz's law, which states that the induced emf will always work in such a direction as to oppose the change in flux that produced it.

This is really at the heart of how transformers work, by using the change in current in one coil to induce a voltage in another.

Every time the current in the primary changes, the flux changes, and this change is mirrored in the secondary, inducing a corresponding emf.
electromotive force (emf)
Electromotive force (emf) is a term used to describe the potential difference that is generated by a source鈥攕uch as a battery or, in this case, a transformer.

Even though it is called a "force," it's actually measured in volts.

In the context of transformers, there are two key emf values to understand:
  • **Primary emf (\(\mathcal{E}_1\))**: This is the back emf that acts within the primary coil as a result of a changing current.
  • **Secondary emf (\(\mathcal{E}_2\))**: This is the emf induced in the secondary coil as a result of the change in magnetic flux caused by the primary coil.
It's essential to remember that the key to efficiently transferring energy from the primary to the secondary coil is the ratio of the number of turns in each coil.

This proportionality determines whether the secondary voltage is higher or lower than the primary. In our example, if you want to transfer more potential energy (voltage), you have more turns in the secondary coil compared to the primary.

This is why transformers are efficient devices for changing the voltage levels of alternating currents.
magnetic flux
Magnetic flux is a concept used to represent the quantity of magnetism, taking into account the strength and the extent of a magnetic field.

Think of it as the number of magnetic field lines passing through a given area.

The magnetic flux \(\Phi\) through a surface is given by the integral of the magnetic field \(B\) over the area \(A\) it penetrates. In formula terms:\[\Phi = \int B \cdot dA\]
  • \(\Phi\) is the magnetic flux
  • \(B\) is the magnetic field
  • \(A\) is the area through which the field lines pass
A change in this magnetic flux over time, especially when the flux is passing through loops of wire, is what creates the induced emf as laid out by Faraday's Law.

Within the transformer setup, having the same flux pass through both the primary and secondary coils ensures that the effect of the changing current is mirrored from the primary coil to the secondary.

This synchronicity is why transformers can efficiently transfer energy between circuits.

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Most popular questions from this chapter

An alternating current \(I_{0} \cos (\omega t)\) (amplitude \(0.5 \mathrm{~A}\), frequency \(60 \mathrm{~Hz}\) ) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius \(1 \mathrm{~cm}\), outer radius \(2 \mathrm{~cm}\), height \(1 \mathrm{~cm}, 1000\) tums). The coil is connected to a 500 \(\Omega\) resistor. (a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, \(I_{r}(t)\). in the resistor. (b) Calculate the back emf in the coil, due to the current \(I_{r}(t) .\) What is the ratio of the amplitudes of this back emf and the "direct" emf in (a)?

A perfectly conducting spherical shell of radius \(a\) rotates about the \(z\) axis with angular velocity \(\omega\), in a uniform magnetic field \(\mathbf{B}=B_{0}\). \(\mathbf{z}\). Calculate the emf developed between the "north pole" and the equator, [Answer: \(\left.\frac{1}{2} B_{0} \omega a^{2}\right]\)

Electrons undergoing cyclotron motion can be speeded up by increasing the magnetic field; the accompanying electric field will impart tangential acceleration. This is the principle of the betatron. One would like to keep the radius of the orbit constant during the process. Show that this can be achieved by designing a magnet such that the average field over the area of the orbit is twice the ficld at the circumference (Fig. 7.52). Assume the electrons start from rest in zcro ficld, and that the apparatus is symmetric about the center of the orbit. (Assume also that the electron velocity remains weil below the speed of light, so that nonrelativistic mechanics applies.)

A square loop (side \(a\) ) is mounted on a vertical shaft and rotated at angular velocity \(\omega\) (Fig. 7.18). A uniform magnetic field \(\mathbf{B}\) points to the right. Find the \(\mathcal{E}(t)\) for this alternating current generator.

A long solenoid with radius \(a\) and \(n\) turns per unit length carries a time- dependent current \(I(t)\) in the \(\hat{\phi}\) direction. Find the electric field (magnitude and direction) at a distance \(s\) from the axis (both inside and outside the solenoid), in the quasistatic approximation.
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