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(a)
The magnitude and direction of induced emf, induced current, and force on the loop.
(a)
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Answer to Problem 89P
The magnitude of induced emf is
Explanation of Solution
The loop is at rest and magnetic field is constant, and average velocity of electron is also zero
Write the expression for magnetic force
Here,
Since velocity of electrons is zero, according to equation (I) the magnetic force will be zero.
Write the expression for induced emf
Here,
From equation (II) when velocity of electrons become zero, the emf is also become zero.
The expression for induced current
Here,
Since emf is zero, induced current is also zero according to equation (I)
Conclusion:
Therefore, the magnitude of induced emf is
(b)
Repeat part (a) for a loop moving to right with constant speed
(b)
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Answer to Problem 89P
The magnitude of induced emf is
Explanation of Solution
In this case electrons are moving through the loop and there is a magnetic field associated with the motion of electrons. According to right hand rule, when a magnetic field is in the direction shown by curled fingers, the direction of force is along the right thumb. Since negatively charged electrons moves to right, the direction of field is downward in the plane of paper.
Write the expression for magnetic field in a loop
Here,
The expression for magnetic force is
The force will develop an emf in loop in the left and right side, since left side is close to the wire, the magnetic field will be greater at right side, hence electron circulate in counterclockwise direction, and current flows in clockwise direction.
Write the expression for induced emf
Here,
Substitute,
The current in the loop is
The magnetic force acting on each segment of loop be
The expression for magnitude of force is
Substitute,
Conclusion:
Substitute,
Substitute,
Substitute,
The force is acting to left.
Therefore, the magnitude of induced emf is
(c)
The electric power dissipated in the loop, and show that it is equal to the rate of external force.
(c)
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Answer to Problem 89P
The electric power dissipated in the loop is
Explanation of Solution
Write the expression for power dissipated in the loop
Here,
Write the expression for power dissipated by external force
Here,
Conclusion:
Substitute,
Substitute,
Thus,
Therefore, the electric power dissipated in the loop is
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