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Chapter 22: Problem 56

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 12.8 A.

Step by step solution

01

Understand the Transformer's Turns Ratio

The transformer's turns ratio is given as 1:8. This means that for every one turn of wire in the primary coil, there are eight turns in the secondary coil. This ratio affects the voltage and current transformation between the primary and secondary coils.
02

Recall the Transformer Current Relationship

For transformers, the relationship between the primary and secondary currents is inversely proportional to the turns ratio. This can be expressed as: \[ \frac{I_p}{I_s} = \frac{N_s}{N_p} \] where \( I_p \) and \( I_s \) are the currents in the primary and secondary coils, respectively, and \( N_p \) and \( N_s \) are the number of turns in the primary and secondary coils, respectively.
03

Insert Known Values Into Formula

We know the secondary current \( I_s \) is 1.6 A and the turns ratio \( \frac{N_p}{N_s} = \frac{1}{8} \). Substituting these values into the formula gives: \[ \frac{I_p}{1.6} = \frac{8}{1} \]
04

Solve for Primary Current \( I_p \)

Rearrange the equation from Step 3 to solve for the primary current \( I_p \) by multiplying both sides with 1.6: \[ I_p = 1.6 imes 8 \] Calculating this gives \( I_p = 12.8 \,\mathrm{A} \).
05

Conclusion: Current in Primary Coil

The current in the primary coil when the train is running is calculated to be 12.8 A.

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Key Concepts

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

Understanding Turns Ratio
A transformer is a device that adjusts voltage levels between its input and output. One of the critical factors in a transformer is the turns ratio, which directly influences voltage and current changes. The turns ratio is simply the ratio of the number of windings (or turns) in the primary coil to the number of turns in the secondary coil.

For example, a turns ratio of 1:8 indicates that for every single turn of wire in the primary coil, there are eight turns in the secondary coil. This ratio plays a fundamental role in determining how voltage and current are adjusted within the transformer.
  • A transformer with a high turns ratio can step down high voltages to lower voltages, which is why the electric train needs a transformer to adjust the wall voltage to a suitable level.
  • Conversely, this affects the current in the opposite way as voltage鈥攊f voltage is decreased, the current is increased, keeping the transformer's power constant (ignoring losses).
Role of Primary and Secondary Coils
In a transformer, energy transformation between the coils is critical. A transformer's primary coil is connected to the input power source, while the secondary coil is connected to the output, providing power to the device needing transformation.

The operation involves:
  • Primary Coil: This coil is responsible for drawing power from the source. It's this part that the number of turns influences how voltage and current are to be transformed.
  • Secondary Coil: This part releases transformed energy to the load (in this exercise, it powers the electric train). With a greater number of turns as given by our step-down transformer example, it decreases the voltage.
The interaction between these coils determines the balance between voltage and current levels, ensuring that the required power levels are correctly delivered to the train.
Explaining Current Transformation
Current transformation determines how current values change between the primary and secondary coils, directly influenced by the turns ratio.

In our exercise, we use the concept of inverse proportionality related to turns ratio. This means that if voltage decreases due to a step-down transformer configuration, the secondary current increases according to the inverse of the turns ratio.

This is mathematically expressed by the equation:\[ \frac{I_p}{I_s} = \frac{N_s}{N_p} \]Here, \( I_p \) and \( I_s \) are primary and secondary currents, and \( N_p \) and \( N_s \) are the number of turns.
  • In our 1:8 turns ratio example, the formula becomes \( \frac{I_p}{1.6} = \frac{8}{1} \), thus showing the primary current will be significantly higher than the secondary.
  • This characteristic ensures essential devices like electric trains receive appropriate power levels for their operation despite the lowered voltage.

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Most popular questions from this chapter

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is \(45.0 \Omega\). The area of each turn is \(4.70 \times 10^{-4} \mathrm{~m}^{2}\). This coil is moved from a region where the magnetic field is zero into a region where it is nonzero, the normal to the coil being kept parallel to the magnetic field. The amount of charge that is induced to flow around the circuit is measured to be \(8.87 \times 10^{-3} \mathrm{C}\). Find the magnitude of the magnetic field. (Such a device can be used to measure the magnetic field strength and is called a flux meter.)

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. (a) Why is an emf induced in the coil? (b) What determines the amount of induced current in the coil? (c) How is the amount of charge \(\Delta q\) that flows related to the induced current \(I\) and the time interval \(t-t_{0}\) during which the coil rotates? 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 when 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?

A \(3.0-\mu F\) capacitor has a voltage of \(35 \mathrm{~V}\) between its plates. What must be the current in a 5.0-mH inductor, such that the energy stored in the inductor equals the energy stored in the capacitor?

A planar coil of wire has a single turn. The normal to this coil is parallel to a uniform and constant (in time) magnetic field of \(1.7 \mathrm{~T}\). An emf that has a magnitude of \(2.6 \mathrm{~V}\) is induced in this coil because the coil's area \(A\) is shrinking. What is the magnitude of \(\Delta A / \Delta t,\) which is the rate \(\left(\right.\) in \(\left.m^{2} / s\right)\) at which the area changes?

The drawing shows a coil of copper wire that consists of two semicircles joined by straight sections of wire. In part \(a\) the coil is lying flat on a horizontal surface. The dashed line also lies in the plane of the horizontal surface. Starting from the orientation in part \(a\), the smaller semicircle rotates at an angular frequency \(\omega\) about the dashed line, until its plane becomes perpendicular to the horizontal surface, as shown in part \(b\). A uniform magnetic field is constant in time and is directed upward, perpendicular to the horizontal surface. The field completely fills the region occupied by the coil in either part of the drawing. (a) In which part of the drawing, if either, does a greater magnetic flux pass through the coil? Account for your answer. (b) As the shape of the coil changes from that in part \(a\) of the drawing to that in part \(b\), does an induced current flow in the coil, and, if so, in which direction does it flow? Give your reasoning. To describe the flow, imagine that you are above the coil looking down at it. (c) How is the period \(T\) of the rotational motion related to the angular frequency \(\omega\), and in terms of the period, what is the shortest time interval that elapses between parts \(a\) and \(b\) of the drawing? the magnitude of the magnetic field is \(0.35 \mathrm{~T}\). The resistance of the coil is \(0.025 \Omega\), and the smaller semicircle has a radius of \(0.20 \mathrm{~m}\). The angular frequency at which the small semicircle rotates is \(1.5 \mathrm{rad} / \mathrm{s}\). Determine the average current, if any, induced in the coil as the coil changes shape from that in part \(a\) of the drawing to that in \(\operatorname{part} b\).
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