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Chapter 22: Problem 52

During a 72 -ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a \(6.0-\mathrm{mA}\) 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 mH. What is the change in the primary current?

Short Answer

Expert verified
The change in the primary current is 1.62 A.

Step by step solution

01

Understanding the Given Information

We are given a secondary coil current of 6.0 mA, a resistance of 12 惟, a mutual inductance of 3.2 mH, and a time interval of 72 ms. We need to find the change in the primary current.
02

Using Ohm's Law to Find Induced Voltage

First, apply Ohm's Law to find the voltage induced in the secondary coil. Ohm's Law states that \( V = IR \), where \( I = 6.0\,\mathrm{mA} = 6.0 \times 10^{-3}\,\mathrm{A} \) and \( R = 12\,\Omega \). Therefore, the induced voltage \( V \) is \( 6.0 \times 10^{-3} \times 12 = 0.072\,\mathrm{V} \).
03

Relating Induced Voltage to Rate of Change of Current

The induced EMF in the secondary coil, \( V \), can be related to the mutual inductance and the rate of current change in the primary coil using the formula \( V = M \frac{dI_p}{dt} \), where \( M = 3.2\,\mathrm{mH} = 3.2 \times 10^{-3}\,\mathrm{H} \).
04

Solving for the Change in Primary Current

We have \( 0.072 = 3.2 \times 10^{-3} \frac{dI_p}{dt} \). Solve for \( \frac{dI_p}{dt} \) by dividing both sides by \( 3.2 \times 10^{-3} \). Hence, \( \frac{dI_p}{dt} = \frac{0.072}{3.2 \times 10^{-3}} = 22.5\,\mathrm{A/s} \).
05

Calculating Total Change in Primary Current

The total change in primary current, \( \Delta I_p \), over the interval \( \Delta t = 72\,\mathrm{ms} = 72 \times 10^{-3}\,\mathrm{s} \) is obtained via the formula \( \Delta I_p = \frac{dI_p}{dt} \times \Delta t \). Substitute \( \frac{dI_p}{dt} = 22.5\,\mathrm{A/s} \) and \( \Delta t = 72 \times 10^{-3} \): \( \Delta I_p = 22.5 \times 72 \times 10^{-3} = 1.62\,\mathrm{A} \).

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Key Concepts

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

Ohm's Law
Ohm's Law is a fundamental principle in electronics and physics that relates voltage, current, and resistance. It is expressed as:
\[ V = IR \]
where \( V \) is the voltage (in volts), \( I \) is the current (in amperes), and \( R \) is the resistance (in ohms).
  • This law illustrates how voltage in a circuit depends on both current and resistance.
  • In our problem, the secondary coil creates a current of 6.0 mA (or \( 6.0 \times 10^{-3} \) A) due to the induced voltage and has a resistance of 12 惟.
  • Using Ohm's Law, the induced voltage \( V \) can be calculated as \( 0.072 \) V after multiplying the current and the resistance.
Ohm's Law helps in understanding the basic relationship between different electrical components, providing the foundation to determine other factors like induced EMF.
Induced EMF
Induced electromotive force (EMF) refers to the voltage generated by changing magnetic fields in a coil. This phenomenon is based on Faraday's Law of Electromagnetic Induction, which states that a change in magnetic flux can induce an EMF.
  • In our scenario, the mutual inductance between the two coils is 3.2 mH, representing the effectiveness of the primary coil's changing current in inducing a voltage in the secondary coil.
  • The relationship is given by \( V = M \frac{dI_p}{dt} \), where \( M \) is the mutual inductance, and \( \frac{dI_p}{dt} \) is the rate of change of the primary coil's current.
The induced EMF in this situation is calculated to be 0.072 V from Ohm's Law, allowing us to relate it back to the change in current in the primary coil. Induced EMF is a vital concept in understanding how transformers and many other electrical devices function.
Primary Coil Current
The primary coil current changes over time, creating a varying magnetic field that influences the secondary coil. Understanding this dynamic is crucial for systems relying on mutual induction.
  • We calculate the change in primary current with the equation \( \Delta I_p = \frac{dI_p}{dt} \times \Delta t \).
  • From the solution, the rate of change \( \frac{dI_p}{dt} \) is found to be 22.5 A/s, and the time interval \( \Delta t \) is 72 milliseconds.
  • By multiplying these values, the total change in primary current \( \Delta I_p \) is 1.62 A.
This change shows how alterations in the primary coil's current can induce EMF and affect the secondary coil. Understanding this concept aids in grasping how devices like transformers and inductors operate effectively.

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Most popular questions from this chapter

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 \(N_{s} / N_{\mathrm{p}}\)

You need to design a \(60.0-\mathrm{Hz}\) ac generator that has a maximum emf of \(5500 \mathrm{V}\). The generator is to contain a 150 -turn coil that has an area per turn of \(0.85 \mathrm{m}^{2} .\) What should be the magnitude of the magnetic field in which the coil rotates?

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?

A circular coil (950 turns, radius \(=0.060 \mathrm{m}\) ) is rotating in a uniform magnetic field. At \(t=0\) s, the normal to the coil is perpendicular to the magnetic field. At \(t=0.010 \mathrm{s}\), the normal makes an angle of \(\phi=45^{\circ}\) with the field because the coil has made one-eighth of a revolution. An average emf of magnitude \(0.065 \mathrm{V}\) is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.

The drawing shows a type of flow meter that can be used to measure the speed of blood in situations when a blood vessel is sufficiently exposed (e.g., during surgery). Blood is conductive enough that it can be treated as a moving conductor. When it flows perpendicularly with respect to a magnetic field, as in the drawing, electrodes can be used to measure the small voltage that develops across the vessel. Suppose that the speed of the blood is \(0.30 \mathrm{m} / \mathrm{s}\) and the diameter of the vessel is \(5.6 \mathrm{mm} .\) In a \(0.60-\mathrm{T}\) magnetic field what is the magnitude of the voltage that is measured with the electrodes in the drawing?
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