(a)
The magnetic flux through the loop due to the current.
(a)
Answer to Problem 31.13P
The magnetic flux through the loop due to the current is
Explanation of Solution
Given info: The length of the rectangle is
The magnetic field is a function of the distance. So, the value of the magnetic field varies over the area of the rectangular loop.
Consider an infinitesimal section of the loop at a distance of
Write the expression for the area of the infinitesimal section of the loop.
Here,
Write the expression for the magnetic field at the infinitesimal section.
Here,
Write the expression for the magnetic flux through the infinitesimal section.
Substitute
Write the expression for the magnetic flux through the entire loop.
Conclusion:
Therefore, the magnetic flux through the loop due to the current is
(b)
The induced emf in the loop.
(b)
Answer to Problem 31.13P
The induced emf in the loop is
Explanation of Solution
Given info: The length of the rectangle is
Write the expression for the induced emf in the loop.
Here,
From part (a), the magnetic flux through the loop due to the current is
Substitute
Substitute
Substitute
Conclusion:
Therefore, the induced emf in the loop is
(c)
The direction of the induced current in the rectangle.
(c)
Answer to Problem 31.13P
The direction of the induced current in the rectangle is counterclockwise.
Explanation of Solution
Given info: The length of the rectangle is
Write the expression for the current induced in the loop.
Here,
From the above expression, the current is directly proportional to the emf induced in the loop. From part (b), the induced emf in the loop is
Conclusion:
Therefore, the direction of the induced current in the rectangle is counterclockwise.
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Chapter 31 Solutions
Physics For Scientists And Engineers, Technology Update, Loose-leaf Version
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