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The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at θ = 0 size 12{θ=0} {} . The torque then reverses its direction once the coil rotates past θ = 0 size 12{θ=0} {} . (See [link] (d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at θ = 0 size 12{θ=0} {} . To get the coil to continue rotating in the same direction, we can reverse the current as it passes through θ = 0 size 12{θ=0} {} with automatic switches called brushes . (See [link] .)

The diagram shows a current-carrying loop between the north and south poles of a magnet at two different times. The north pole is to the left and the south pole is to the right. The magnetic field runs from the north pole to the right to the south pole. Figure a shows the current running through the loop. It runs up on the left side, and down on the right side. The force on the left side is into the page. The force on the right side is out of the page. The torque is clockwise when viewed from above. Figure b shows the loop when it is oriented perpendicular to the magnet. In both diagrams, the bottom of each side of the loop is connected to a half-cylinder that is next to a rectangular brush that is then connected to the rest of the circuit.
(a) As the angular momentum of the coil carries it through θ = 0 size 12{θ=0} {} , the brushes reverse the current to keep the torque clockwise. (b) The coil will rotate continuously in the clockwise direction, with the current reversing each half revolution to maintain the clockwise torque.

Meters , such as those in analog fuel gauges on a car, are another common application of magnetic torque on a current-carrying loop. [link] shows that a meter is very similar in construction to a motor. The meter in the figure has its magnets shaped to limit the effect of θ size 12{θ} {} by making B size 12{B} {} perpendicular to the loop over a large angular range. Thus the torque is proportional to I size 12{I} {} and not θ size 12{θ} {} . A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection proportional to I size 12{I} {} . If an exact proportionality cannot be achieved, the gauge reading can be calibrated. To produce a galvanometer for use in analog voltmeters and ammeters that have a low resistance and respond to small currents, we use a large loop area A size 12{A} {} , high magnetic field B size 12{B} {} , and low-resistance coils.

Diagram of a meter showing a current-carrying loop between two poles of a magnet. The torque on the magnet is clockwise. The top of the loop is connected to a spring and to a pointer that points to a scale as the loop rotates.
Meters are very similar to motors but only rotate through a part of a revolution. The magnetic poles of this meter are shaped to keep the component of B size 12{B} {} perpendicular to the loop constant, so that the torque does not depend on θ size 12{θ} {} and the deflection against the return spring is proportional only to the current I size 12{I} {} .

Section summary

  • The torque τ size 12{τ} {} on a current-carrying loop of any shape in a uniform magnetic field. is
    τ = NIAB sin θ , size 12{τ= ital "NIAB""sin"θ} {}
    where N size 12{N} {} is the number of turns, I size 12{I} {} is the current, A size 12{A} {} is the area of the loop, B size 12{B} {} is the magnetic field strength, and θ size 12{θ} {} is the angle between the perpendicular to the loop and the magnetic field.

Conceptual questions

Draw a diagram and use RHR-1 to show that the forces on the top and bottom segments of the motor’s current loop in [link] are vertical and produce no torque about the axis of rotation.

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(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) τ size 12{" τ"} {} decreases by 5.00% if B decreases by 5.00%

(b) 5.26% increase

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(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 size 12{θ} {} is 10 . 9º? size 12{"10" "." 9°?} {}

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Find the current through a loop needed to create a maximum torque of 9 . 00 N m. size 12{9 "." "00"`N cdot m "." } {} 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

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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 size 12{3"00"`N cdot m} {} if the loop is carrying 25.0 A.

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Since the equation for torque on a current-carrying loop is τ = NIAB sin θ size 12{τ= ital "NIAB""sin"θ} {} , the units of N m size 12{N cdot m} {} must equal units of A m 2 T size 12{A cdot m rSup { size 8{2} } `T} {} . Verify this.

A m 2 T = A m 2 N A m = N m size 12{A cdot m rSup { size 8{2} } cdot T=A cdot m rSup { size 8{2} } left ( { {N} over {A cdot m} } right )=N cdot m} {} .

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(a) At what angle θ size 12{θ} {} is the torque on a current loop 90.0% of maximum? (b) 50.0% of maximum? (c) 10.0% of maximum?

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A proton has a magnetic field due to its spin on its axis. The field is similar to that created by a circular current loop 0 . 650 × 10 15 m size 12{0 "." "650" times "10" rSup { size 8{ - "15"} } `m} {} in radius with a current of 1 . 05 × 10 4 A size 12{1 "." "05" times "10" rSup { size 8{4} } `A} {} (no kidding). Find the maximum torque on a proton in a 2.50-T field. (This is a significant torque on a small particle.)

3 . 48 × 10 26 N m size 12{3 "." "48" times "10" rSup { size 8{ - "26"} } `N cdot m} {}

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(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. The Earth’s field here is due north, parallel to the ground, with a strength of 3 . 00 × 10 5 T size 12{3 "." "00" times "10" rSup { size 8{ - 5} } `T} {} . What are the direction and magnitude of the torque on the loop? (b) Does this device have any practical applications as a motor?

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Repeat [link] , but with the loop lying flat on the ground with its current circulating counterclockwise (when viewed from above) in a location where the Earth’s field is north, but at an angle 45 . size 12{"45" "." 0°} {} below the horizontal and with a strength of 6. 00 × 10 5 T size 12{6 "." "00" times "10" rSup { size 8{ - 5} } `T} {} .

(a) 0.666 N m size 12{0 "." "666"`N cdot m} {} west

(b) This is not a very significant torque, so practical use would be limited. Also, the current would need to be alternated to make the loop rotate (otherwise it would oscillate).

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Questions & Answers

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Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Practice Key Terms 2

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