# 22.8 Torque on a current loop: motors and meters

 Page 1 / 4
• Describe how motors and meters work in terms of torque on a current loop.
• Calculate the torque on a current-carrying loop in a magnetic field.

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 [link] .)

Let us examine the force on each segment of the loop in [link] 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 $w$ and height $l$ . 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. [link] shows views of the loop from above. Torque is defined as $\tau =\text{rF}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ , where $F$ is the force, $r$ is the distance from the pivot that the force is applied, and $\theta$ is the angle between $r$ and $F$ . As seen in [link] (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 $r=w/2$ , the torque on each vertical segment is $\left(w/2\right)F\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ , and the two add to give a total torque.

$\tau =\frac{w}{2}F\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta +\frac{w}{2}F\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{wF}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$

Now, each vertical segment has a length $l$ that is perpendicular to $B$ , so that the force on each is $F=\text{IlB}$ . Entering $F$ into the expression for torque yields

$\tau =\text{wIlB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta .$

If we have a multiple loop of $N$ turns, we get $N$ times the torque of one loop. Finally, note that the area of the loop is $A=\text{wl}$ ; the expression for the torque becomes

$\tau =\text{NIAB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta .$

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 $I$ , has $N$ turns, each of area $A$ , and the perpendicular to the loop makes an angle $\theta$ with the field $B$ . The net force on the loop is zero.

## 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 $\tau =\text{NIAB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta$ . Maximum torque occurs when $\theta =\text{90º}$ and $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =1$ .

Solution

For $\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =1$ , the maximum torque is

${\tau }_{\text{max}}=\text{NIAB}.$

Entering known values yields

$\begin{array}{lll}{\tau }_{\text{max}}& =& \left(\text{100}\right)\left(\text{15.0 A}\right)\left(\text{0.100}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\right)\left(2\text{.}\text{00 T}\right)\\ & =& \text{30.0 N}\cdot m.\end{array}$

Discussion

This torque is large enough to be useful in a motor.

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
What is thermal heat all about
why uniform circular motion is called a periodic motion?.
when a train start from A & it returns at same station A . what is its acceleration?
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
what are the types of radioactivity
Worku
what is static friction
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
Boniface
the range of objects and phenomena studied in physics is
Boniface