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To find the net torque on the current loop shown in [link] , we first consider and Since they have the same line of action and are equal and opposite, the sum of their torques about any axis is zero (see Fixed-Axis Rotation ). Thus, if there is any torque on the loop, it must be furnished by and Let’s calculate the torques around the axis that passes through point O of [link] (a side view of the coil) and is perpendicular to the plane of the page. The point O is a distance x from side 2 and a distance from side 4 of the loop. The moment arms of and are and respectively, so the net torque on the loop is
This simplifies to
where is the area of the loop.
Notice that this torque is independent of x ; it is therefore independent of where point O is located in the plane of the current loop. Consequently, the loop experiences the same torque from the magnetic field about any axis in the plane of the loop and parallel to the x -axis.
A closed-current loop is commonly referred to as a magnetic dipole and the term IA is known as its magnetic dipole moment Actually, the magnetic dipole moment is a vector that is defined as
where is a unit vector directed perpendicular to the plane of the loop (see [link] ). The direction of is obtained with the RHR-2—if you curl the fingers of your right hand in the direction of current flow in the loop, then your thumb points along If the loop contains N turns of wire, then its magnetic dipole moment is given by
In terms of the magnetic dipole moment, the torque on a current loop due to a uniform magnetic field can be written simply as
This equation holds for a current loop in a two-dimensional plane of arbitrary shape.
Using a calculation analogous to that found in Capacitance for an electric dipole, the potential energy of a magnetic dipole is
Check Your Understanding
In what orientation would a magnetic dipole have to be to produce (a) a maximum torque in a magnetic field? (b) A maximum energy of the dipole?
a. aligned or anti-aligned; b. perpendicular
(a) By how many percent is the torque of a motor decreased if its permanent magnets lose 5.0% of their strength? (b) How many percent would the current need to be increased to return the torque to original values?
a. so decreases by 5.00% if B decreases by 5.00%; b. 5.26% increase
(a) What is the maximum torque on a 150-turn square loop of wire 18.0 cm on a side that carries a 50.0-A current in a 1.60-T field? (b) What is the torque when θ is 10.9º?
Find the current through a loop needed to create a maximum torque of The loop has 50 square turns that are 15.0 cm on a side and is in a uniform 0.800-T magnetic field.
10.0 A
Calculate the magnetic field strength needed on a 200-turn square loop 20.0 cm on a side to create a maximum torque of 300 N ⋅ m if the loop is carrying 25.0 A.
Since the equation for torque on a current-carrying loop is τ = NIAB sin θ , the units of N ⋅ m must equal units of A ⋅ m 2 T. Verify this.
(a) At what angle θ is the torque on a current loop 90.0% of maximum? (b) 50.0% of maximum? (c) 10.0% of maximum?
A proton has a magnetic field due to its spin. The field is similar to that created by a circular current loop in radius with a current of Find the maximum torque on a proton in a 2.50-T field. (This is a significant torque on a small particle.)
(a) A 200-turn circular loop of radius 50.0 cm is vertical, with its axis on an east-west line. A current of 100 A circulates clockwise in the loop when viewed from the east. Earth’s field here is due north, parallel to the ground, with a strength of What are the direction and magnitude of the torque on the loop? (b) Does this device have any practical applications as a motor?
Repeat the previous problem, but with the loop lying flat on the ground with its current circulating counterclockwise (when viewed from above) in a location where Earth’s field is north, but at an angle 45.0° below the horizontal and with a strength of
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