# 13.3 Motional emf  (Page 4/6)

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## Significance

An alternative way of looking at the induced emf from Faraday’s law is to integrate in space instead of time. The solution, however, would be the same. The motional emf is

$|\epsilon |=\int Bvdl.$

The velocity can be written as the angular velocity times the radius and the differential length written as dr . Therefore,

$|\epsilon |=B\int vdr=B\omega \underset{0}{\overset{l}{\int }}rdr=\frac{1}{2}B\omega {l}^{2},$

which is the same solution as before.

## A rectangular coil rotating in a magnetic field

A rectangular coil of area A and N turns is placed in a uniform magnetic field $\stackrel{\to }{B}=B\stackrel{^}{j},$ as shown in [link] . The coil is rotated about the z -axis through its center at a constant angular velocity $\omega .$ Obtain an expression for the induced emf in the coil.

## Strategy

According to the diagram, the angle between the perpendicular to the surface ( $\stackrel{^}{n}$ ) and the magnetic field $\left(\stackrel{\to }{B}\right)$ is $\theta$ . The dot product of $\stackrel{\to }{B}·\stackrel{^}{n}$ simplifies to only the $\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ component of the magnetic field, namely where the magnetic field projects onto the unit area vector $\stackrel{^}{n}$ . The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify quickly. The induced emf is written out using Faraday’s law.

## Solution

When the coil is in a position such that its normal vector $\stackrel{^}{n}$ makes an angle $\theta$ with the magnetic field $\stackrel{\to }{B},$ the magnetic flux through a single turn of the coil is

${\text{Φ}}_{\text{m}}={\int }_{S}\stackrel{\to }{B}·\stackrel{^}{n}dA=BA\text{cos}\phantom{\rule{0.2em}{0ex}}\theta .$

From Faraday’s law, the emf induced in the coil is

$\epsilon =\text{−}N\frac{d{\text{Φ}}_{\text{m}}}{dt}=NBA\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \frac{d\theta }{dt}.$

The constant angular velocity is $\omega =d\theta \text{/}dt.$ The angle $\theta$ represents the time evolution of the angular velocity or $\omega t$ . This is changes the function to time space rather than $\theta$ . The induced emf therefore varies sinusoidally with time according to

$\epsilon ={\epsilon }_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\omega t,$

where ${\epsilon }_{0}=NBA\omega .$

## Significance

If the magnetic field strength or area of the loop were also changing over time, these variables wouldn’t be able to be pulled out of the time derivative to simply the solution as shown. This example is the basis for an electric generator, as we will give a full discussion in Applications of Newton’s Law .

Check Your Understanding Shown below is a rod of length l that is rotated counterclockwise around the axis through O by the torque due to $m\stackrel{\to }{g}.$ Assuming that the rod is in a uniform magnetic field $\stackrel{\to }{B}$ , what is the emf induced between the ends of the rod when its angular velocity is $\omega$ ? Which end of the rod is at a higher potential?

$\epsilon =B{l}^{2}\omega \text{/}2,$ with O at a higher potential than S

Check Your Understanding A rod of length 10 cm moves at a speed of 10 m/s perpendicularly through a 1.5-T magnetic field. What is the potential difference between the ends of the rod?

1.5 V

## Summary

• The relationship between an induced emf $\epsilon$ in a wire moving at a constant speed v through a magnetic field B is given by $\epsilon =Blv.$
• An induced emf from Faraday’s law is created from a motional emf that opposes the change in flux.

## Conceptual questions

A bar magnet falls under the influence of gravity along the axis of a long copper tube. If air resistance is negligible, will there be a force to oppose the descent of the magnet? If so, will the magnet reach a terminal velocity?

What is differential form of Gauss's law?
help me out on this question the permittivity of diamond is 1.46*10^-10.( a)what is the dielectric of diamond (b) what its susceptibility
a body is projected vertically upward of 30kmp/h how long will it take to reach a point 0.5km bellow e point of projection
i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
kya ye b.sc ka hai agar haa to konsa part
what is charge quantization
it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
it is just like bohr's theory. Which was angular momentum of electron is intral multiple of h/2π
determine absolute zero
The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
what is copper loss
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
what is the weight of the earth in space
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
explain the lack of symmetry in the field of the parallel capacitor
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe