<< Chapter < Page Chapter >> Page >
  • 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] .)

Diagram showing a current-carrying loop of width w and length l between the north and south poles of a magnet. The north pole is to the left and the south pole is to the right of the loop. The magnetic field B runs from the north pole across the loop to the south pole. The loop is shown at an instant, while rotating clockwise. The current runs up the left side of the loop, across the top, and down the right side. There is a force F oriented into the page on the left side of the loop and a force F oriented out of the page on the right side of the loop. The torque on the loop is clockwise as viewed from above.
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 [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 τ = rF sin θ size 12{τ= ital "rF""sin"θ} {} , where F size 12{F} {} is the force, r is the distance from the pivot that the force is applied, and θ 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 ( w / 2 ) F sin θ , and the two add to give a total torque.

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} {}
Diagram showing a current-carrying loop from the top, and four different times as it rotates in a magnetic field. The magnetic field oriented toward the right, perpendicular to the vertical dimension of the loop. In figure a, the top view of the loop is oriented at an angle to the magnetic field lines, which run left to right. The force on the loop is up on the lower left side where the current comes out of the page. The force is down on the upper right side where the loop goes into the page. The angle between the force and the loop is theta. Torque is clockwise and equals w over 2 times I l B sine theta. Figure b shows the top view of the loop parallel to the magnetic field lines. The force on the loop is up on the left side where I comes out of the page. The force on the loop is down on the right side where I goes into the page. The angle theta between the F and B is ninety degrees. Torque is clockwise and equals w over 2 I l B equals maximum torque. Figure c shows the top view of the loop oriented perpendicular to B. The force on the loop is up at the top, where I comes out of the page, and down at the bottom where I goes into the page. Theta equals 0 degrees. Torque equals zero since sine theta equals 0. In figure d the force is down on the lower left side of the loop where I goes in, and up on the upper right side of the loop where I comes out. The torque is counterclockwise. Torque is negative.
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 θ size 12{θ} {} with the field that is the same as the angle between w / 2 size 12{w/2} {} and F size 12{F} {} . (b) The maximum torque occurs when θ size 12{θ} {} is a right angle and sin θ = 1 size 12{"sin"θ=1} {} . (c) Zero (minimum) torque occurs when θ size 12{θ} {} is zero and sin θ = 0 . (d) The torque reverses once the loop rotates past θ = 0 .

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

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

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

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

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 size 12{I} {} , has N size 12{N} {} turns, each of area A size 12{A} {} , and the perpendicular to the loop makes an angle θ size 12{θ} {} with the field B size 12{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 τ = NIAB sin θ size 12{τ= ital "NIAB""sin"θ} {} . Maximum torque occurs when θ = 90º and sin θ = 1 size 12{"sin"θ=1} {} .

Solution

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

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

Entering known values yields

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 "." {} } } {}

Discussion

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

Got questions? Get instant answers now!

Questions & Answers

What is the difference between a principle and a law
the law is universally proved. The principal depends on certain conditions.
Dr
state Faraday first law
aliyu Reply
it states that mass of an element deposited during electrolysis is directly proportional to the quantity of electricity discharge
Olamide
what does the speedometer of a car measure ?
Jyoti Reply
Car speedometer measures the rate of change of distance per unit time.
Moses
describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air)
WILLIAM Reply
using the law of reflection explain how powder takes the shine off a person's nose. what is the name of the optical effect?
WILLIAM
is higher resolution of microscope using red or blue light?.explain
WILLIAM
what is dimensional consistent
Mohammed
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed
syed
can sound wave in air be polarized?
WILLIAM Reply
Unlike transverse waves such as electromagnetic waves, longitudinal waves such as sound waves cannot be polarized. ... Since sound waves vibrate along their direction of propagation, they cannot be polarized
Astronomy
A proton moves at 7.50×107m/s perpendicular to a magnetic field. The field causes the proton to travel in a circular path of radius 0.800 m. What is the field strength?
Celedonio Reply
derived dimenionsal formula
Ajak Reply
what is the difference between mass and weight
Isru Reply
assume that a boy was born when his father was eighteen years.if the boy is thirteen years old now, how is his father in
Isru
31yrs
Olamide
what is head-on collision
Javaid Reply
what is airflow
Godswill Reply
derivative of first differential equation
Haruna Reply
why static friction is greater than Kinetic friction
Ali Reply
draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
nkombo Reply
What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
what is energy
Yusuf
How electric lines and equipotential surface are mutually perpendicular?
Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask