Concept explainers
A rectangular loop of wire has dimensions 0.500 m by 0.300 m. The loop is pivoted at the x axis and lies in the xy plane as shown in Figure P28.34. A uniform magnetic field of magnitude 1.50 T is directed at an angle of 40.0° with respect to the y axis with field lines parallel to the yz plane. The loop carries a current of 0.900 A in the direction shown. (Ignore gravitation.) We wish to evaluate the torque on the current loop. (a) What is the direction of the magnetic force exerted on wire segment ab? (b) What is the direction of the torque associated with this force about an axis through the origin? (c) What is the direction of the magnetic force exerted on segment cd? (d) What is the direction of the torque associated with this force about an axis through the origin? (e) Can the forces examined in parts (a) and (c) combine to cause the Loop to rotate around the x axis? (f) Can they affect the motion of the loop in any way? Explain. (g) What is the direction of the magnetic force exerted on segment bc? (h) What is the direction of the torque associated with this force about an axis through the origin? (i) What is the torque on segment ad about an axis through the origin? (j) From the point of view of Figure P28.34, once the loop is released from rest at the position shown, will it rotate clockwise or counterclockwise around the x axis? (k) Compute the magnitude of the magnetic moment of the loop. (l) What is the angle between the magnetic moment sector and the magnetic field? (m) Compute the torque on the loop using the results to parts (k) and (l).
Figure P28.34
(a)
The direction of the magnetic force exerted on wire segment
Answer to Problem 34P
The direction of the force is in the positive x-axis.
Explanation of Solution
Given Info: The length of the rectangular loop is
According to the Fleming left hand rule, if a thumb, an index finger and a middle finger of a right hand is stretch in a position that they formed mutually perpendicular to each other, there the direction of thumb will indicate the direction of force, the direction of index finger will indicate the direction of magnetic field and the direction of middle finger will indicate the direction of current.
By using the Fleming left hand rule, if the middle finger shows the direction of current in segment
Conclusion:
Therefore, the direction of the force is in the positive x-axis.
(b)
The direction of the torque associated with the above force about an axis through the origin.
Answer to Problem 34P
The direction of the torque associated with the force on segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the torque associated with the force is,
Here,
Substitute
The torque associated in the above expression in the negative of the unit vector
Conclusion:
Therefore, the direction of the torque associated with the force on segment
(c)
The direction of the magnetic force exerted on the segment
Answer to Problem 34P
The direction of the magnetic force exerted on the segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the magnetic force on the wire segment
Here,
Substitute
In the above expression it is shown that the magnetic force is in the negative of the unit vector
Conclusion:
Therefore, the direction of the magnetic force exerted on the segment
(d)
The direction of the torque associated with the force on the segment
Answer to Problem 34P
The direction of the torque associated with the force on segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the torque associated with the force is,
Here,
Substitute
The torque associated in the above expression is in the negative of the unit vector
Conclusion:
Therefore, the direction of the torque associated with the force on segment
(e)
Whether the force examined in part (a) and (c) combine to cause the loop to rotate around the x-axis.
Answer to Problem 34P
The magnetic force cannot rotate the loop.
Explanation of Solution
Given Info: The length of the rectangular loop is
As the force examined on the segment
Thus the net force on the rectangular loop become zero.
Conclusion:
Therefore, the magnetic force cannot rotate the loop.
(f)
Whether the above forces affect the motion of the loop.
Answer to Problem 34P
The magnetic force will only rotate the loop and will not affect the motion of the loop.
Explanation of Solution
Given Info: The length of the rectangular loop is
According to the expression in the equation (2) if the magnetic field, current and the length of the loop is constant then its magnetic force will be constant too. In the given problem the magnetic field, the current and the length of the loop is constant, therefore, the magnetic force is constant. Hence this magnetic force will only rotate the loop and will not affect the motion of the loop.
Conclusion:
Therefore, the magnetic force will only rotate the loop and will not affect the motion of the loop.
(g)
The direction of the magnetic force exerted on segment
Answer to Problem 34P
The magnetic force on segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the magnetic force on the wire segment
Here,
Substitute
In the above expression it is shown that the magnetic force is in the negative of the unit vector
Conclusion:
Therefore, the magnetic force on segment
(h)
The direction of the torque associated with the force on the segment
Answer to Problem 34P
The direction of the torque associated with the force on segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the torque associated with the force is,
Here,
Substitute
The torque associated in the above expression is in the positive of the unit vector
Conclusion:
Therefore, the direction of the torque associated with the force on segment
(i)
The direction of the torque on the segment
Answer to Problem 34P
The torque on the segment
Explanation of Solution
Given Info: The length of the rectangular loop is
The Formula to calculate the torque associated with the force is,
The segment
Conclusion:
Therefore, the torque on the segment
(j)
Whether the loop locate itself clockwise or anticlockwise.
Answer to Problem 34P
The rectangular loop will rotate in the anticlockwise direction.
Explanation of Solution
Given Info: The length of the rectangular loop is
As the torque on the segment
Conclusion:
Therefore, the rectangular loop will rotate in the anticlockwise direction.
(k)
The magnitude of the magnetic moment of the loop.
Answer to Problem 34P
The magnitude of the magnetic moment of the loop is
Explanation of Solution
Given Info: The length of the rectangular loop is
The formula to calculate the magnetic moment of the loop is,
Here,
Substitute
Conclusion:
Therefore, the magnitude of the magnetic moment of the loop is
(l)
The angle between the magnetic moment vector and magnetic field.
Answer to Problem 34P
The angle between the magnetic moment and magnetic field is
Explanation of Solution
Given Info: The length of the rectangular loop is
It is given that the current if lowing in clockwise direction therefore, here follow the principle of right hand thumb rule, if the finger is curl in the direction of current then the thumb will indicate the moment of magnetic field. Thus, the direction of magnetic moment is downward. That is along the negative z direction. The angle between the magnetic moment and magnetic field is,
Conclusion:
Therefore, the angle between the magnetic moment and magnetic field is
(m)
The torque on the loop using the results of part (k) and (l).
Answer to Problem 34P
The torque on the loop is
Explanation of Solution
Given Info: The length of the rectangular loop is
Formula to calculate the torque in current carrying wire is,
Substitute
Conclusion:
Therefore, the torque on the loop is
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