Torque on a Current Loop: Motors and Meters
Now that we have figured out how the magnetic force on a straight wire depends on the magnetic field and the current, we can now bend the wire into a loop, or coil. This gives rise to two new applications:
Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process.
Meters, such as the galvanometer, are another common application of magnetic torque on a current-carrying loop. This finally answers the question how ammeters actually work (we treated them as black boxes in the discussion of circuits earlier). As with motors, the basic idea is to convert the magnetic force into a twisting action, also called torque. This is how the indicator arrow on a meter is made to rotate to the appropriate position, indicating how many amperes are flowing through the meter.
Read this text, which discusses these applications.
Motors are the most common application of magnetic force on current-carrying wires. Motors have loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts torque on the loops, which rotates
a shaft. Electrical energy is converted to mechanical work in the process. (See Figure 22.34.)
Figure 22.34 Torque on a current loop. A current-carrying loop of wire attached to a vertically rotating shaft feels magnetic forces that produce a clockwise torque as viewed from above.
Let us examine the force on each segment of the loop in Figure 22.34 to find the torques produced about the axis of the vertical shaft. (This will lead to a useful equation for the torque on the loop.) We take the magnetic field to
be uniform over the rectangular loop, which has width and height
. First, we note that the forces on the top and bottom segments are vertical and, therefore, parallel to the shaft, producing no torque.
Those vertical forces are equal in magnitude and opposite in direction, so that they also produce no net force on the loop. Figure 22.35 shows views of the loop from above. Torque is defined as , where
is
the force,
is the distance from the pivot that the force is applied, and
is the angle between
and
. As seen in
Figure 22.35(a), right hand rule 1 gives the forces on the sides to be equal in magnitude and opposite in direction, so that the net force is again zero. However, each force produces a clockwise torque. Since
, the torque
on each vertical segment is
, and the two add to give a total torque.
Figure 22.35 Top views of a current-carrying loop in a magnetic field. (a) The equation for torque is derived using this view. Note that the perpendicular to the loop makes an angle with the field that is the same as the angle between
and
. (b) The maximum torque occurs when
is a right angle and
. (c) Zero (minimum) torque occurs when
is zero and
. (d) The torque reverses once the loop rotates past
.
Now, each vertical segment has a length that is perpendicular to
, so that the force on each is
. Entering
into the expression for torque yields
If we have a multiple loop of turns, we get
times the torque of one loop. Finally, note that the area of the loop is
; the expression for the torque becomes
This is the torque on a current-carrying loop in a uniform magnetic field. This equation can be shown to be valid for a loop of any shape. The loop carries a current , has
turns, each of area
, and the perpendicular to the loop makes an
angle
with the field
. The net force on the loop is zero.
Example 22.5 Calculating Torque on a Current-Carrying Loop in a Strong Magnetic Field
Find the maximum torque on a 100-turn square loop of a wire of 10.0 cm on a side that carries 15.0 A of current in a 2.00-T field.
Strategy
Torque on the loop can be found using . Maximum torque occurs when
and
.
Solution
Entering known values yields
Discussion
This torque is large enough to be useful in a motor.
The torque found in the preceding example is the maximum. As the coil rotates, the torque decreases to zero at . The torque then reverses its direction once the coil rotates past
. (See Figure 22.35(d).)
This means that, unless we do something, the coil will oscillate back and forth about equilibrium at
. To get the coil to continue rotating in the same direction, we can reverse the current as it passes through
with automatic
switches called brushes. (See Figure 22.36.)
Figure 22.36 (a) As the angular momentum of the coil carries it through , 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. Figure 22.37 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 by making
perpendicular to the loop over a large angular range. Thus the torque is proportional to
and not
. A linear spring exerts a counter-torque that balances the current-produced
torque. This makes the needle deflection proportional to
. 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
, high magnetic field
, and low-resistance coils.
Figure 22.37 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 perpendicular to the loop constant, so that the torque does not depend on
and the deflection against the return spring is proportional only to the current
.
Source: Rice University, https://openstax.org/books/college-physics/pages/22-8-torque-on-a-current-loop-motors-and-meters This work is licensed under a Creative Commons Attribution 4.0 License.