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Chapter 22: Problem 65

A step-down transformer (turns ratio \(=1: 8\) ) is used with an electric train to reduce the voltage from the wall receptacle to a value needed to operate the train. When the train is running, the current in the secondary coil is \(1.6 \mathrm{A}\). What is the current in the primary coil?

Short Answer

Expert verified
The current in the primary coil is 0.2 A.

Step by step solution

01

Understanding the Transformer Turns Ratio

In a transformer, the turns ratio, denoted as \(n\), is the ratio between the number of turns in the primary coil (\(N_p\)) and the secondary coil (\(N_s\)). Given in the problem, the turns ratio is \(1:8\), which means \(N_p:N_s = 1:8\) or \(n = \frac{1}{8}\).
02

Applying the Transformer Current Relationship

In a transformer, the current ratio is the inverse of the turns ratio. If \(I_p\) is the current in the primary coil and \(I_s\) is the current in the secondary coil, the relation is \[ I_p \times N_p = I_s \times N_s \]. Given \(I_s = 1.6 \mathrm{A}\) and \(N_s = 8N_p\), the current relation becomes \[ I_p = \frac{I_s}{n} \].
03

Calculating the Primary Current

Using the current relationship formula from the previous step, substitute the given values: \[ I_p = \frac{1.6 \mathrm{A}}{8} \]. So, the primary current \(I_p\) is \(0.2 \mathrm{A}\).

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Key Concepts

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

Understanding Step-Down Transformers
A step-down transformer is a device that reduces high voltage to a lower voltage. This is vital in various applications, especially with devices that cannot handle high voltages, like an electric train.
  • Primary Coil: This is where the high voltage enters.
  • Secondary Coil: This is where the lower voltage exits.
Inside the transformer, these coils are wrapped around a core. By changing the number of turns in the coils, transformers can adjust the voltage levels. This capability is essential for safely running low-voltage equipment on standard power sources, such as those from a wall socket.
Exploring Turns Ratio
The turns ratio is a key factor in determining how a transformer's voltage changes. It's the ratio of turns between the primary and secondary coils.
For example, a turns ratio of \(1:8\) means the primary coil has one turn for every eight in the secondary coil. This directly influences how voltage is stepped down.
  • Higher ratio: More voltage reduction.
  • Lower ratio: Less voltage reduction.
Understanding the turns ratio is crucial for predicting how much the voltage will change, helping us determine what kind of voltage will be supplied to devices.
Delving into Current Transformation
In a transformer, the relationship between current and the turns ratio is straightforward. The current in the secondary and primary coils is inversely proportional to the turns ratio.
  • Current in Primary Coil \( (I_p) \): Calculated using the inverse of the turns ratio.
  • Current in Secondary Coil \( (I_s) \): Given or measurable from devices.
For our example with a turns ratio of \(1:8\), the equation \( I_p = \frac{I_s}{n} \) helps you find the primary current. If \( I_s = 1.6 \mathrm{A} \), then \( I_p = 0.2 \mathrm{A} \). This shows how effectively a transformer can safely modify current for different applications.
Electric Train Voltage Reduction
Electric trains often need lower voltages than what standard power outlets provide. A step-down transformer is the solution, ensuring safe and efficient power use.
  • Step-down transformers modify voltage to meet train requirements.
  • They protect sensitive components from high voltage damage.
By converting standard power outlet voltages to a train-compatible level, transformers enable trains to operate efficiently, while maintaining safety and durability of the train's electronic systems.

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Most popular questions from this chapter

Consult Multiple-Concept Example 11 for background material relating to this problem. A small rubber wheel on the shaft of a bicycle generator presses against the bike tire and turns the coil of the generator at an angular speed that is 38 times as great as the angular speed of the tire itself. Each tire has a radius of \(0.300 \mathrm{m}\). The coil consists of 125 turns, has an area of \(3.86 \times 10^{-3} \mathrm{m}^{2},\) and rotates in a \(0.0900-\mathrm{T}\) magnetic field. The bicycle starts from rest and has an acceleration of \(+0.550 \mathrm{m} / \mathrm{s}^{2} .\) What is the peak emf produced by the generator at the end of 5.10 s?

The plane of a flat, circular loop of wire is horizontal. An external magnetic field is directed perpendicular to the plane of the loop. The magnitude of the external magnetic field is increasing with time. Because of this increasing magnetic field, an induced current is flowing clockwise in the loop, as viewed from above. What is the direction of the external magnetic field? Justify your conclusion.

The secondary coil of a step-up transformer provides the voltage that operates an electrostatic air filter. The turns ratio of the transformer is \(50: 1 .\) The primary coil is plugged into a standard \(120-\mathrm{V}\) outlet. The current in the secondary coil is \(1.7 \times 10^{-3} \mathrm{A}\). Find the power consumed by the air filter.

A piece of copper wire is formed into a single circular loop of radius \(12 \mathrm{cm} .\) A magnetic field is oriented parallel to the normal to the loop. and it increases from 0 to \(0.60 \mathrm{T}\) in a time of \(0.45 \mathrm{s}\). The wire has a resistance per unit length of \(3.3 \times 10^{-2} \Omega / \mathrm{m} .\) What is the average electrical energy dissipated in the resistance of the wire?

A constant magnetic field passes through a single rectangular loop whose dimensions are \(0.35 \mathrm{m} \times 0.55 \mathrm{m} .\) The magnetic field has a magnitude of \(2.1 \mathrm{T}\) and is inclined at an angle of \(65^{\circ}\) with respect to the normal to the plane of the loop. (a) If the magnetic field decreases to zero in a time of \(0.45 \mathrm{s},\) what is the magnitude of the average emf induced in the loop? (b) If the magnetic field remains constant at its initial value of \(2.1 \mathrm{T},\) what is the magnitude of the rate \(\Delta A / \Delta t\) at which the area should change so that the average emf has the same magnitude as in part (a)?
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