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Concept explainers
(a)
The direction of the magnetic force on each of the four sides of the rectangle due to the long wire’s magnetic field.
(a)
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Answer to Problem 108P
The direction of magnetic force each side of the rectangle is given in below table.
Side | Current direction | Field direction | Force direction |
Top | right | Out of the page | Down: attracted to long wire |
Bottom | Left | Out of the page | Up: repelled by long wire |
Left | Up | Out of the page | Right |
Right | down | Out of the page | left |
Explanation of Solution
Write the expression for the magnitude of magnetic field due to current carrying long wire.
Here,
The direction of magnetic field is given by right hand rule. According to the rule, when the thumb is pointed in the direction of current and the fingers are curled, the direction of fingers represents the direction of magnetic field lines. The tangent of the field line at any point gives the direction of magnetic field at that point.
Since current flows to right, magnetic field points out of the page.
Write the expression for force acting on one side of current carrying rectangular loop of wire.
Here,
The direction of force is given by the direction of
Consider the bottom side of rectangular loop, where current is flowing to the left and magnetic field is out of the page. According to right hand rule, magnetic force acts in the upward direction. That is repelled by long wire.
Consider the top side of rectangular loop, where current is flowing to the right and magnetic field is out of the page. According to right hand rule, magnetic force acts in the downward direction. That is attracted to the long wire.
Consider the left side of rectangular loop, where current is flowing up and magnetic field is out of the page. According to right hand rule, magnetic force acts towards the right.
Consider the right side of rectangular loop, where current is flowing downward and magnetic field is out of the page. According to right hand rule, magnetic force acts toward the left.
Conclusion:
Therefore, the direction of magnetic force each side of the rectangle is given in below table.
Side |
Current direction | Field direction | Force direction |
Top | right | Out of the page | Down: attracted to long wire |
Bottom | Left | Out of the page | Up: repelled by long wire |
Left | Up | Out of the page | Right |
Right | down | Out of the page | left |
(b)
The net magnetic force on the rectangular loop due to the long wire’s magnetic field.
(b)
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Answer to Problem 108P
The net magnetic force on the rectangular loop due to the long wire’s magnetic field is
Explanation of Solution
The magnetic field along the left and right side of the rectangular loop have same magnitude at each point of wire. The left and right side of the rectangular loop experience equal magnitude of magnetic force, since the two sides are symmetrically situated with respected to long wire. The top of the loop experiences small magnetic field than bottom side, since radial distance of top side is larger than that of the bottom side.
Since magnetic forces on left and right side of the loop are equal in magnitude and opposite in direction, they cancel each other.
Write the expression for the net force acting on the rectangular loop.
Here,
The negative sign indicates that force on top and bottom side are opposite in direction.
Write the expression to calculate magnitude of magnetic force on each side of wire.
Here,
From equation (I), write the expression for the magnitude of magnetic force on bottom wire.
Here,
From equation (I), write the expression for the magnitude of magnetic force on top wire.
Here,
Write the expression for
Here,
Write the expression for
Here,
Substitute
Here,
Substitute
Substitute
Conclusion:
Substitute
Since answer is positive, the net force must direct along the direction of force acting on the top side of rectangular loop. Therefore, net force is directed away from the long wire.
Therefore, the net magnetic force on the rectangular loop due to the long wire’s magnetic field is
(c)
The magnetic force on the long wire due to the loop.
(c)
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Answer to Problem 108P
The magnetic force on the long wire due to the loop is
Explanation of Solution
According to
The net magnetic force on the rectangular loop due to the long wire’s magnetic field is
Conclusion:
The net magnetic force on the rectangular loop due to the long wire’s magnetic field is
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Chapter 19 Solutions
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