/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 The resistances of the primary a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 22: Problem 63

The resistances of the primary and secondary coils of a transformer are 56 and \(14 \Omega,\) respectively. Both coils are made from lengths of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio \(N_{s} / N_{\mathrm{p}}\)

Short Answer

Expert verified
The turns ratio \(N_{s}/N_{p}\) is 1/4.

Step by step solution

01

Understand the Relationship of Resistances to Number of Turns

The resistances of the primary and secondary coils are given as 56 惟 and 14 惟, respectively. Because both coils are made from the same copper wire and have the same diameter, their resistances are proportional to their number of turns. Thus, we can write:\[ \frac{R_{p}}{R_{s}} = \frac{N_{p}}{N_{s}} \] where \(R_{p}\) is the resistance of the primary coil, \(R_{s}\) is the resistance of the secondary coil, \(N_{p}\) is the number of turns in the primary coil, and \(N_{s}\) is the number of turns in the secondary coil.
02

Substitute Known Values Into the Equation

Substitute the given resistance values into the equation from Step 1:\[ \frac{56}{14} = \frac{N_{p}}{N_{s}} \]
03

Simplify the Equation

Simplify the fraction on the left side of the equation:\[ 4 = \frac{N_{p}}{N_{s}} \] This simplifies further to show that the turns ratio \(N_{s}/N_{p}\) is the reciprocal of 4.
04

Solve for the Turns Ratio

Taking the reciprocal of both sides, we get:\[ \frac{N_{s}}{N_{p}} = \frac{1}{4} \] Thus, the turns ratio is 1/4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding the Turns Ratio
In transformers, the turns ratio is a crucial concept that helps determine how voltage is transformed from the primary coil to the secondary coil. It's defined as the ratio of the number of turns in the secondary coil ( _{s} ) to the number of turns in the primary coil ( _{p} ). When you know this ratio, you can predict how the voltage will change as it passes through the transformer.

If you have more turns on the secondary coil compared to the primary coil, the output voltage will be higher, which is called step-up transformation. Conversely, if the primary coil has more turns, the voltage is reduced, known as step-down transformation.
  • The equation for the turns ratio is given by ( _{s} / _{p} ).
  • This ratio is helpful in designing transformers to meet specific voltage requirements.
In our exercise, because the primary coil's resistance is 56 惟 and the secondary's is 14 惟 , their turns ratio is ( _{s} / _{p} = 1/4 ). This means the transformer reduces voltage to a quarter of the original value.
The Role of Electrical Resistance
Electrical resistance plays a significant role in determining the efficiency and effectiveness of a transformer. It is defined as the opposition to current flow within an electrical circuit. In the context of transformers, both the primary and secondary coils have intrinsic resistances, which impact how current moves through them.

Resistance ( R ) is associated with the material and dimensions of the wire:
  • Since both coils in the exercise are made from the same copper wire, they share similar resistivity factors.
  • Resistance can be calculated with the formula ( R = 蟻 ( l/A ), where 蟻 is the resistivity, l is length, and A is the cross-sectional area.
The resistance of the coil directly influences the number of turns because longer wire (more turns) means a higher resistance, assuming cross-sectional characteristics remain constant.

In this problem, we see a direct proportionality between resistance and turns, hence the inverse relation in the formula for turns ratio.
Primary and Secondary Coils of a Transformer
Transformers consist of two essential components: the primary and secondary coils. These coils are intricately wound wires that play distinct roles in the transformer鈥檚 operation.

- **Primary Coil:** This is the coil connected to the input voltage source. It's where electrical energy is initially supplied. - **Secondary Coil:** This coil is connected to the load or where the transformed voltage is delivered.

The interaction between these coils allows transformers to modify voltage levels based on the turns ratio. When alternating current passes through the primary coil, it creates a magnetic field, and this field induces a current in the secondary coil through electromagnetic induction.
  • The efficiency of voltage transformation depends significantly on the construction and properties of these coils.
  • For instance, having the same wire diameter, as in the exercise, ensures uniform energy transfer characteristics.
Understanding these transformations and the design of coils helps in building efficient transformers for varied applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A magnetic field has a magnitude of \(0.078 \mathrm{T}\) and is uniform over a circular surface whose radius is \(0.10 \mathrm{m}\). The field is oriented at an angle of \(\phi=25^{\circ}\) with respect to the normal to the surface. What is the magnetic flux through the surface?

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R\). It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of \(140 \Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5} \mathrm{C}\) flows in the coil. What is the magnitude of the magnetic field?

Suppose there are two transformers between your house and the high-voltage transmission line that distributes the power. In addition, assume that your house is the only one using electric power. At a substation the primary coil of a step-down transformer (turns ratio \(=1: 29\) ) receives the voltage from the high-voltage transmission line. Because of your usage, a current of \(48 \mathrm{mA}\) exists in the primary coil of this transformer. The secondary coil is connected to the primary of another step-down transformer (turns ratio \(=1: 32\) ) somewhere near your house, perhaps up on a telephone pole. The secondary coil of this transformer delivers a \(240-\mathrm{V}\) emf to your house. How much power is your house using? Remember that the current and voltage given in this problem are rms values.

A circular loop of wire rests on a table. A long, straight wire lies on this loop, directly over its center, as the drawing illustrates. The current \(I\) in the straight wire is decreasing. In what direction is the induced current, if any, in the loop? Give your reasoning.

Coil 1 is a flat circular coil that has \(N_{1}\) turns and a radius \(R_{1}\). At its center is a much smaller flat, circular coil that has \(N_{2}\) turns and radius \(R_{2}\). The planes of the coils are parallel. Assume that coil 2 is so small that the magnetic field due to coil 1 has nearly the same value at all points covered by the area of coil \(2 .\) Determine an expression for the mutual inductance between these two coils in terms of \(\mu_{0}, N_{1}, R_{1}, N_{2},\) and \(R_{2}\)
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Translational Dynamics

Read Explanation

Fundamentals of Physics

Read Explanation

Physics of Motion

Read Explanation

Electromagnetism

Read Explanation

Atoms and Radioactivity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.