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Chapter 22: Problem 54

The batteries in a portable CD player are recharged by a unit that plugs into a wall socket. Inside the unit is a step-down transformer with a turns ratio of \(1: 13 .\) The wall socket provides \(120 \mathrm{~V}\). What voltage does the secondary coil of the transformer provide?

Short Answer

Expert verified
The secondary coil provides approximately 9.23 V.

Step by step solution

01

Understanding Transformer Basics

A transformer is a device that changes the voltage levels of an alternating current (AC) supply using two sets of coils: primary and secondary. The ratio of the number of turns in the primary coil to the number of turns in the secondary coil (turns ratio) determines the step-up or step-down voltage transformation.
02

Identifying Provided Information

In this problem, the transformer is a step-down type with a turns ratio of \(1:13\). This means for every turn in the primary coil, there are 13 turns in the secondary coil. The primary voltage provided by the wall socket is \(120 \text{ V}\).
03

Using the Turns Ratio

The turns ratio is given as \(1:13\), which indicates that the primary voltage to the secondary voltage is in the ratio of \(1:13\). Thus using the formula for transformers, \( \frac{V_p}{V_s} = \frac{N_p}{N_s} \), where \(V_p\) and \(V_s\) are the primary and secondary voltages respectively, and \(N_p\) and \(N_s\) are the turns on the primary and secondary coil respectively.
04

Calculating Secondary Voltage

Given \(V_p = 120 \text{ V}\) and the turns ratio \( \frac{N_p}{N_s} = \frac{1}{13} \), the secondary voltage \(V_s\) can be found by rearranging the formula: \( V_s = V_p \times \frac{N_s}{N_p} = 120 \times \frac{1}{13} \). This simplifies to \( V_s = \frac{120}{13} \approx 9.23 \text{ V}\).

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Key Concepts

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

Step-Down Transformer
A step-down transformer is a special electrical device which reduces or "steps down" the voltage level of an alternating current when it passes through. It consists of two winding coils: the primary and the secondary coil. The primary coil is connected to the main power source, usually wall socket voltage, while the secondary coil delivers the reduced voltage output that is suitable for specific devices.

Here's why they are widely used:
  • They provide the correct voltage levels needed by smaller electronic devices, protecting them from high voltages.
  • They exceptionally maintain efficiency and reliability, ensuring minimal energy loss during transformation.
In our exercise, the step-down transformer makes sure that the high wall socket voltage of 120 V is adjusted to a safer, lower level of 9.23 V suitable for charging a CD player.
Turns Ratio
The turns ratio is a fundamental aspect of transformers that defines the relationship between the number of turns in its primary and secondary coils. It's represented as a simple fraction: the number of coil turns in the primary to that in the secondary. In simple words, it dictates how much your input voltage will get transformed.

To calculate the turns ratio:
  • Let the primary turns be denoted as \(N_p\) and the secondary turns as \(N_s\).
  • The ratio of primary to secondary coil turns \(\frac{N_p}{N_s}\) directly specifies voltage transformation.
The given turns ratio of 1:13 in our problem implies that for every one turn of wire in the primary coil, there are thirteen turns in the secondary coil. This effectively reduces the input voltage, resulting in a lower output voltage.
Secondary Voltage
Secondary voltage refers to the altered voltage output delivered by the secondary coil of a transformer. This conversion process retains the alternating nature of current but adjusts its amplitude.

The secondary voltage \(V_s\) can be found using the formula: \[V_s = V_p \times \frac{N_s}{N_p}\]where \(V_p\) is the primary voltage. For our exercise, substituting the given values results in:\[V_s = 120 \times \frac{1}{13} \approx 9.23\text{ V}\]This calculated secondary voltage is essential to power electronic devices effectively at safe levels without causing damage.
Alternating Current
Alternating current (AC) is the type of electrical current employed by transformers and most household appliances. Unlike direct current (DC), which flows in a single direction, AC reverses its direction periodically. This cyclical flow makes it compatible and ideal for transformers due to its ease in converting voltages over long distances without efficiency loss.

AC's properties are:
  • It frequently changes direction at a standardized frequency, typically 50 or 60 Hz depending on your region.
  • Transformers, including step-down models, utilize these properties to adjust the voltage of the current to desired levels effectively.
This reliance on alternating current enables devices like a CD player to receive a safe and suitable power supply after transformation.

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Most popular questions from this chapter

A vacuum cleaner is plugged into a \(120.0\) -V socket and uses \(3.0\) A of current in normal operation when the back emf generated by the electric motor is \(72.0 \mathrm{~V}\). Find the coil resistance of the motor.

The earth's magnetic field, like any magnetic field, stores energy. The maximum strength of the earth's field is about \(7.0 \times 10^{-5} \mathrm{~T}\). Find the maximum magnetic energy stored in the space above a city if the space occupies an area of \(5.0 \times 10^{8} \mathrm{~m}^{2}\) and has a height of \(1500 \mathrm{~m}\).

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?

During a 72 -ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a \(6.0\) -mA current in a nearby secondary coil. The secondary coil is part of a circuit in which the resistance is \(12 \Omega\). The mutual inductance between the two coils is \(3.2 \mathrm{mH}\). What is the change in the primary current?

A \(120.0\) - \(\mathrm{V}\) motor draws a current of \(7.00 \mathrm{~A}\) when running at normal speed. The resistance of the armature wire is \(0.720 \Omega\). (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to \(15.0 \mathrm{~A} ?\)
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