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Chapter 22: Problem 33

ssm 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.

Short Answer

Expert verified
The external magnetic field is directed upwards.

Step by step solution

01

Understand Faraday's Law of Induction

Faraday's law states that a change in magnetic flux through a loop induces an electromotive force (EMF) and hence a current in the loop. The direction of the induced current is such that it opposes the change in magnetic flux (Lenz's Law).
02

Determine the Relationship Between the Magnetic Field and Induced Current

Since the induced current is clockwise when viewed from above, according to the right-hand rule, the induced magnetic field created by this current must be pointing downwards, opposing the increase in the external magnetic field.
03

Apply Lenz's Law to Identify the Direction of the External Magnetic Field

To oppose the increase in the external magnetic field, the direction of the external magnetic field must be opposite to the direction of the induced magnetic field. Therefore, since the induced magnetic field is downwards, the external magnetic field must be upwards to cause an increasing flux in the downward direction.

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Key Concepts

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

Faraday's Law
Faraday's Law is a fundamental principle that describes how electric currents are generated by changing magnetic fields. To put it simply:
  • Whenever the magnetic flux through a loop changes, an electromotive force (\(EMF\)) is induced.
  • This \(EMF\) generates a current if there is a closed conducting path.
In the context of our exercise, as the external magnetic field increases, it changes the magnetic flux through the loop. This results in an induced current flowing clockwise. Faraday's Law can be mathematically expressed as \(EMF = -\frac{d\Phi}{dt}\), where \(\Phi\) is the magnetic flux. The negative sign indicates the direction of the induced \(EMF\) as stipulated by Lenz's Law, which we'll talk about next. Understanding this relationship is crucial for determining the behavior of the current and the subsequent magnetic interactions.
Lenz's Law
Lenz's Law is a principle that complements Faraday's Law. It tells us the specific direction that the induced current takes. According to Lenz's Law:
  • The direction of the induced current is always such that it opposes the change in magnetic flux that produced it.
For our scenario, as the external magnetic field's magnitude increases, the loop induces a current to counteract that increase. When the induced current flows clockwise, it indicates that the induced magnetic field is downward. Lenz's Law is embodied in Faraday's equation by the negative sign, symbolizing this opposition. Understanding Lenz's Law helps in predicting how the system will react to changes and is vital to determining the direction and nature of both currents and magnetic fields.
Magnetic Flux
Magnetic flux is a measure of the magnetic field passing through a given area. It's like counting how many magnetic field lines cross through a loop. To visualize it:
  • Imagine a magnetic field as invisible lines passing through the loop.
  • The greater the number of lines, the greater the flux.
The flux (\(\Phi\)) is calculated by the formula \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(B\) is the magnetic field strength, \(A\) is the area of the loop, and \(\theta\) is the angle between the field lines and the perpendicular to the loop's surface. In this context, since the loop is perpendicular to the magnetic field, \(\theta\) is zero and \(\cos(0)=1\), which simplifies the flux calculation to \(\Phi = B \cdot A\). This concept is key when applying Faraday's Law to understand the induced \(EMF\).
Right-hand Rule
The right-hand rule is a simple way to determine the direction of a magnetic field or current. To use it to find the direction of the magnetic field created by the current:
  • Point your thumb in the direction of the conventional current (from positive to negative).
  • Your curled fingers show the direction of the magnetic field loops around the wire.
In our scenario, since the current is clockwise when viewed from above, the right-hand rule indicates that the magnetic field due to this induced current is directed downward. This rule is essential, especially when analyzing interactions between the magnetic field created by currents and the external magnetic field. By understanding and applying the right-hand rule, the behavior of magnetic fields and their impacts becomes more accessible.

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Most popular questions from this chapter

In some places, insect "zappers," with their blue lights, are a familiar sight on a summer's night. These devices use a high voltage to electrocute insects. One such device uses an ac voltage of 4320 \(\mathrm{V}\) , which is obtained from a standard \(120.0-\mathrm{V}\) outlet by means of a transformer. If the primary coil has 21 turns, how many turns are in the secondary coil?

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