3.8 Torque on a current loop: motors and meters (Page 2/4)

Basic physics for medical imaging Page 2 / 4

The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at $θ = 0$ . The torque then reverses its direction once the coil rotates past $θ = 0$ . (See [link] (d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at $θ = 0$ . To get the coil to continue rotating in the same direction, we can reverse the current as it passes through $θ = 0$ 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 $θ = 0$ , 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 $θ$ by making $B$ perpendicular to the loop over a large angular range. Thus the torque is proportional to $I$ and not $θ$ . A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection proportional to $I$ . 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 $A$ , high magnetic field $B$ , 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 $B$ perpendicular to the loop constant, so that the torque does not depend on $θ$ and the deflection against the return spring is proportional only to the current $I$ .

Section summary

The torque $τ$ on a current-carrying loop of any shape in a uniform magnetic field. is
$τ = NIAB sin θ,$
where $N$ is the number of turns, $I$ is the current, $A$ is the area of the loop, $B$ is the magnetic field strength, and $θ$ 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.

Problems&Exercises

(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) $τ$ 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 $9 . 00 N \cdot m.$ 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

Calculate the magnetic field strength needed on a 200-turn square loop 20.0 cm on a side to create a maximum torque of $300 N \cdot m$ 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 $N \cdot m$ must equal units of $A \cdot m^{2} T$ . Verify this.

$A \cdot m^{2} \cdot T = A \cdot m^{2} (\frac{N}{A \cdot m}) = N \cdot m$ .

(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 on its axis. The field is similar to that created by a circular current loop $0 . 650 \times 10^{- 15} m$ in radius with a current of $1 . 05 \times 10^{4} A$ (no kidding). Find the maximum torque on a proton in a 2.50-T field. (This is a significant torque on a small particle.)

$3 . 48 \times 10^{- 26} N \cdot m$

(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 $3 . 00 \times 10^{- 5} T$ . What are the direction and magnitude of the torque on the loop? (b) Does this device have any practical applications as a motor?

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 $45 . 0º$ below the horizontal and with a strength of $6. 00 \times 10^{- 5} T$ .

(a) $0.666 N \cdot m$ 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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Source: OpenStax, Basic physics for medical imaging. OpenStax CNX. Feb 17, 2014 Download for free at http://legacy.cnx.org/content/col11630/1.1

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