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Chapter 6: Problem 7

A horizontal straight wire \(10 \mathrm{~m}\) long extending from east to west is falling with a speed of \(5.0 \mathrm{~m} \mathrm{~s}^{-1}\), at right angles to the horizontal component of the earth's magnetic field, \(0.30 \times 10^{-4} \mathrm{~Wb} \mathrm{~m}^{-2}\). (a) What is the instantaneous value of the emf induced in the wire? (b) What is the direction of the emf? (c) Which end of the wire is at the higher electrical potential?

Short Answer

Expert verified
(a) The instantaneous value of the emf induced in the wire is \(0.015 \mathrm{~V}\) or \(15 \mathrm{~mV}\). (b) The direction of the induced emf is from west to east. (c) The west end of the wire is at a higher electrical potential.

Step by step solution

01

Calculation of the induced emf

We use the equation for the emf induced in a conductor moving in a magnetic field which is defined as \(E = B \cdot L \cdot v\). Where \(E\) is the emf, \(B\) is the magnetic field strength, \(L\) is the length of the wire, and \(v\) is the velocity of the wire. Substituting the given values we get \(E = 0.30 \times 10^{-4} \mathrm{~Wb m}^{-2} \times 10 \mathrm{~m} \times 5.0 \mathrm{~m s}^{-1} = 0.015 \mathrm{~V}\). Therefore, the instantaneous value of the emf induced in the wire is \(0.015 \mathrm{~V}\) or \(15 \mathrm{~mV}\).
02

Direction of the induced emf

To find the direction of the induced emf, we use the right-hand rule and Lenz's Law. The wire is moving downwards, the magnetic field is directed from south to north (in the plane of the paper), so the thumb (direction of emf) of the right hand will point from west to east. Therefore, the direction of the induced emf is from west to east.
03

Determining the end with higher electric potential

The west end of the wire is at a higher electrical potential because the direction of emf is from west to east. This means that electrons in the wire are being pushed from east to west, leaving the west end with a deficit of electrons, hence a higher electrical potential.

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Key Concepts

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

Understanding Magnetic Field
The magnetic field is an invisible force field that surrounds magnetic materials and moving charged particles. It's a core concept in electromagnetism.
A magnetic field can be visualized as lines of force that exert a force on other magnetic materials or charged particles in their vicinity.
Key aspects of magnetic fields include:
  • Magnitude and direction: Represented with vector quantities, meaning they have both strength and direction.
  • Interaction with charged particles: They can exert force on a moving charge or a current-carrying wire.
  • Representation: Often depicted with field lines; closer lines signify a stronger field.
Magnetic fields play a crucial role in inducing electromotive force (emf) in wires when they move through fields at angles, as seen in scenarios where wires move perpendicularly to Earth's magnetic field.
Understanding Lenz's Law
Lenz's Law provides insight into the behavior of induced currents when a conductor, like a wire, moves in a magnetic field.
It states that the direction of the induced current is such that it opposes the change that created it.
According to this principle:
  • Any increase in magnetic flux will generate a current opposing that increase.
  • Conversely, a decrease in magnetic flux will generate a current that attempts to preserve it.
Lenz's Law is applied together with the right-hand rule. Here, it helps determine the direction of the current and hence the direction of the induced emf in scenarios like the wire moving through Earth's magnetic field.
Understanding the Right-Hand Rule
The right-hand rule is a simple and handy method to determine the direction of magnetic forces and current flow in a conductor.
It's essential for understanding how and where the emf is induced in a wire moving through a magnetic field.
Here's how to use it:
  • Point your thumb in the direction of the wire's velocity.
  • Point your fingers in the direction of the magnetic field lines.
  • Your palm faces in the direction of the induced force or emf.
Using this rule helps us find out that the emf is directed from west to east when a wire falls downwards through a magnetic field directed from south to north.
Understanding Electrical Potential
Electrical potential is the measure of potential energy per unit charge at a point in a field. It's a concept closely tied to voltage.
When a charge moves within an electric field, it experiences changes in electrical potential.
Key points about electrical potential:
  • It's about energy: The potential reflects how much work is needed to move a charge.
  • Direction of flow: Higher electrical potential means electrons tend to flow towards it, often leading to current flow.
  • In wires: The end with fewer electrons (and thus more positive) tends to have a higher potential.
In our exercise, the west end of the wire moving through a magnetic field exhibits higher potential, as the induced emf drives electrons eastward, leaving a positive charge on the west.

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Most popular questions from this chapter

A square loop of side \(12 \mathrm{~cm}\) with its sides parallel to \(\mathrm{X}\) and \(\mathrm{Y}\) axes is moved with a velocity of \(8 \mathrm{~cm} \mathrm{~s}^{-1}\) in the positive \(x\) -direction in an environment containing a magnetic field in the positive \(z\) -direction. The field is neither uniform in space nor constant in time. It has a gradient of \(10^{-3} \mathrm{~T} \mathrm{~cm}^{-1}\) along the negative \(x\) -direction (that is it increases by \(10^{-3} \mathrm{~T} \mathrm{~cm}^{-1}\) as one moves in the negative \(x\) -direction), and it is decreasing in time at the rate of \(10^{-3} \mathrm{~T} \mathrm{~s}^{-1} .\) Determine the direction and magnitude of the induced current in the loop if its resistance is \(4.50 \mathrm{~m} \Omega\).

A rectangular wire loop of sides \(8 \mathrm{~cm}\) and \(2 \mathrm{~cm}\) with a small cut is moving out of a region of uniform magnetic field of magnitude \(0.3 \mathrm{~T}\) directed normal to the loop. What is the emf developed across the cut if the velocity of the loop is \(1 \mathrm{~cm} \mathrm{~s}^{-1}\) in a direction normal to the (a) longer side, (b) shorter side of the loop? For how long does the induced voltage last in each case?

A circular coil of radius \(8.0 \mathrm{~cm}\) and 20 turns is rotated about its vertical diameter with an angular speed of \(50 \mathrm{rad} \mathrm{s}^{-1}\) in a uniform horizontal magnetic field of magnitude \(3.0 \times 10^{-2} \mathrm{~T}\). Obtain the maximum and average emf induced in the coil. If the coil forms a closed loop of resistance \(10 \Omega\), calculate the maximum value of current in the coil. Calculate the average power loss due to Joule heating. Where does this power come from?

Figure \(6.20\) shows a metal rod \(\mathrm{PQ}\) resting on the smooth rails \(\mathrm{AB}\) and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch \(\mathrm{K}\). Length of the \(\operatorname{rod}=15 \mathrm{~cm}, B=0.50 \mathrm{~T}\), resistance of the closed loop containing the \(\operatorname{rod}=9.0 \mathrm{~m} \Omega\). Assume the field to be uniform. (a) Suppose \(\mathrm{K}\) is open and the rod is moved with a speed of \(12 \mathrm{~cm} \mathrm{~s}^{-1}\) in the direction shown. Give the polarity and magnitude of the induced emf. (b) Is there an excess charge built up at the ends of the rods when \(\mathrm{K}\) is open? What if \(\mathrm{K}\) is closed? (c) With \(\mathrm{K}\) open and the rod moving uniformly, there is no net force on the electrons in the rod \(\mathrm{PQ}\) even though they do experience magnetic force due to the motion of the rod. Explain. (d) What is the retarding force on the rod when \(\mathrm{K}\) is closed? (e) How much power is required (by an external agent) to keep the rod moving at the same speed \(\left(=12 \mathrm{~cm} \mathrm{~s}^{-1}\right)\) when \(\mathrm{K}\) is closed? How much power is required when \(\mathrm{K}\) is open? (f) How much power is dissipated as heat in the closed circuit? What is the source of this power? (g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

A pair of adjacent coils has a mutual inductance of \(1.5 \mathrm{H} .\) If the current in one coil changes from 0 to \(20 \mathrm{~A}\) in \(0.5 \mathrm{~s}\), what is the change of flux linkage with the other coil?
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