 College Physics

# 22.8Torque on a Current Loop: Motors and Meters

College Physics22.8 Torque on a Current Loop: Motors and Meters

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 $ww$ and height $ll$. 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 $τ=rFsinθτ=rFsinθ size 12{τ= ital "rF""sin"θ} {}$, where $FF size 12{F} {}$ is the force, $rr$ is the distance from the pivot that the force is applied, and $θθ$ is the angle between $rr$ and $FF$. 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 $r=w/2r=w/2$, the torque on each vertical segment is $(w/2)Fsinθ(w/2)Fsinθ$, and the two add to give a total torque.

$τ = w 2 F sin θ + w 2 F sin θ = wF sin θ τ = w 2 F sin θ + w 2 F sin θ = wF sin θ size 12{τ= { {w} over {2} } F"sin"θ+ { {w} over {2} } F"sin"θ= ital "wF""sin"θ} {}$
22.19
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 $θθ size 12{θ} {}$ with the field that is the same as the angle between $w/2w/2 size 12{w/2} {}$ and $FF size 12{F} {}$. (b) The maximum torque occurs when $θθ size 12{θ} {}$ is a right angle and $sinθ=1sinθ=1 size 12{"sin"θ=1} {}$. (c) Zero (minimum) torque occurs when $θθ size 12{θ} {}$ is zero and $sinθ=0sinθ=0$. (d) The torque reverses once the loop rotates past $θ=0θ=0$.

Now, each vertical segment has a length $ll size 12{l} {}$ that is perpendicular to $BB size 12{B} {}$, so that the force on each is $F=IlBF=IlB size 12{F= ital "IlB"} {}$. Entering $FF size 12{F} {}$ into the expression for torque yields

$τ=wIlBsinθ.τ=wIlBsinθ. size 12{τ= ital "wIlB""sin"θ} {}$
22.20

If we have a multiple loop of $NN size 12{N} {}$ turns, we get $NN size 12{N} {}$ times the torque of one loop. Finally, note that the area of the loop is $A=wlA=wl size 12{A= ital "wl"} {}$; the expression for the torque becomes

$τ=NIABsinθ.τ=NIABsinθ. size 12{τ= ital "NIAB""sin"θ} {}$
22.21

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 $II size 12{I} {}$, has $NN size 12{N} {}$ turns, each of area $AA size 12{A} {}$, and the perpendicular to the loop makes an angle $θθ size 12{θ} {}$ with the field $BB size 12{B} {}$. 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 $τ=NIABsinθτ=NIABsinθ size 12{τ= ital "NIAB""sin"θ} {}$. Maximum torque occurs when $θ=90ºθ=90º$ and $sinθ=1sinθ=1 size 12{"sin"θ=1} {}$.

#### Solution

For $sinθ=1sinθ=1 size 12{"sin"θ=1} {}$, the maximum torque is

$τmax=NIAB.τmax=NIAB. size 12{τ rSub { size 8{"max"} } = ital "NIAB"} {}$
22.22

Entering known values yields

τ max = 100 15.0 A 0.100 m 2 2 . 00 T = 30.0 N ⋅ m. τ max = 100 15.0 A 0.100 m 2 2 . 00 T = 30.0 N ⋅ m. alignl { stack { size 12{τ rSub { size 8{"max"} } = left ("100" right ) left ("15" "." 0" A" right ) left (0 "." "100"" m" rSup { size 8{2} } right ) left (2 "." "00"" T" right )} {} # " "="30" "." "0 N" cdot m "." {} } } {}
22.23

#### 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 $θ=0θ=0 size 12{θ=0} {}$. The torque then reverses its direction once the coil rotates past $θ=0θ=0 size 12{θ=0} {}$. (See Figure 22.35(d).) This means that, unless we do something, the coil will oscillate back and forth about equilibrium at $θ=0θ=0 size 12{θ=0} {}$. To get the coil to continue rotating in the same direction, we can reverse the current as it passes through $θ=0θ=0 size 12{θ=0} {}$ with automatic switches called brushes. (See Figure 22.36.)

Figure 22.36 (a) As the angular momentum of the coil carries it through $θ=0θ=0 size 12{θ=0} {}$, 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. 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 $θθ size 12{θ} {}$ by making $BB size 12{B} {}$ perpendicular to the loop over a large angular range. Thus the torque is proportional to $II size 12{I} {}$ and not $θθ size 12{θ} {}$. A linear spring exerts a counter-torque that balances the current-produced torque. This makes the needle deflection proportional to $II size 12{I} {}$. 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 $AA size 12{A} {}$, high magnetic field $BB size 12{B} {}$, 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 $BB size 12{B} {}$ perpendicular to the loop constant, so that the torque does not depend on $θθ size 12{θ} {}$ and the deflection against the return spring is proportional only to the current $II size 12{I} {}$.
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