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A charge of 4.0 μC is distributed uniformly around a thin ring of insulating material. The ring has a radius of 0.20 m and rotates at 2.0 × 10 4 rev/min around the axis that passes through its center and is perpendicular to the plane of the ring. What is the magnetic field at the center of the ring?

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A thin, nonconducting disk of radius R is free to rotate around the axis that passes through its center and is perpendicular to the face of the disk. The disk is charged uniformly with a total charge q . If the disk rotates at a constant angular velocity ω , what is the magnetic field at its center?

B = μ 0 σ ω 2 R

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Consider the disk in the previous problem. Calculate the magnetic field at a point on its central axis that is a distance y above the disk.

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Consider the axial magnetic field B v = μ 0 I R 2 / 2 ( y 2 + R 2 ) 3 / 2 of the circular current loop shown below. (a) Evaluate a a B y d y . Also show that lim a a a B y d y = μ 0 I . (b) Can you deduce this limit without evaluating the integral? ( Hint: See the accompanying figure.)

This picture shows the circular current loop I with the magnetic field B perpendicular to the plane of the loop.

derivation

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The current density in the long, cylindrical wire shown in the accompanying figure varies with distance r from the center of the wire according to J = c r , where c is a constant. (a) What is the current through the wire? (b) What is the magnetic field produced by this current for r R ? For r R ?

This figure shows a long, straight, cylindrical wire with a radius R that has current I flowing through it.
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A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by a from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by J = J 0 k ^ , where J 0 is a constant and k ^ is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R 1 and current density J onto the field of a solid cylindrical conductor with radius R 2 and current density J . Then use the fact that the appropriate azimuthal unit vectors can be expressed as θ ^ 1 = k ^ × r ^ 1 and θ ^ 2 = k ^ × r ^ 2 to show that everywhere inside the cavity the magnetic field is given by the constant B = 1 2 μ 0 J 0 k × a , where a = r 1 r 2 and r 1 = r 1 r ^ 1 is the position of P relative to the center of the conductor and r 2 = r 2 r ^ 2 is the position of P relative to the center of the cavity.

This figure shows a large circle with a radius R1 that has a circular hole of radius R2 in it at a distance a from the center. Point P is located in a hole at the distance r2 from the center of a hole and at a distance r1 from the center of a large circle.

derivation

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Between the two ends of a horseshoe magnet the field is uniform as shown in the diagram. As you move out to outside edges, the field bends. Show by Ampère’s law that the field must bend and thereby the field weakens due to these bends.

This figure shows a horse shoe magnet with the magnetic lines going from the North end to the South end.
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Show that the magnetic field of a thin wire and that of a current loop are zero if you are infinitely far away.

As the radial distance goes to infinity, the magnetic fields of each of these formulae go to zero.

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An Ampère loop is chosen as shown by dashed lines for a parallel constant magnetic field as shown by solid arrows. Calculate B · d l for each side of the loop then find the entire B · d l . Can you think of an Ampère loop that would make the problem easier? Do those results match these?

This figure shows an Ampere loop that is located in the constant magnetic field. One of the sides of the loop forms an angle theta with the magnetic line.
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A very long, thick cylindrical wire of radius R carries a current density J that varies across its cross-section. The magnitude of the current density at a point a distance r from the center of the wire is given by J = J 0 r R , where J 0 is a constant. Find the magnetic field (a) at a point outside the wire and (b) at a point inside the wire. Write your answer in terms of the net current I through the wire.

a. B = μ 0 I 2 π r ; b. B = μ 0 J 0 r 2 3 R

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A very long, cylindrical wire of radius a has a circular hole of radius b in it at a distance d from the center. The wire carries a uniform current of magnitude I through it. The direction of the current in the figure is out of the paper. Find the magnetic field (a) at a point at the edge of the hole closest to the center of the thick wire, (b) at an arbitrary point inside the hole, and (c) at an arbitrary point outside the wire. ( Hint: Think of the hole as a sum of two wires carrying current in the opposite directions.)

This figure shows a circle with a radius a that has a circular hole of radius b in it at a distance d from the center.
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Magnetic field inside a torus. Consider a torus of rectangular cross-section with inner radius a and outer radius b . N turns of an insulated thin wire are wound evenly on the torus tightly all around the torus and connected to a battery producing a steady current I in the wire. Assume that the current on the top and bottom surfaces in the figure is radial, and the current on the inner and outer radii surfaces is vertical. Find the magnetic field inside the torus as a function of radial distance r from the axis.

B ( r ) = μ 0 N I / 2 π r
This figure shows a torus with the inner radius a and an outer radius b. A thin wire is wound evenly on the torus.

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Two long coaxial copper tubes, each of length L , are connected to a battery of voltage V . The inner tube has inner radius a and outer radius b , and the outer tube has inner radius c and outer radius d . The tubes are then disconnected from the battery and rotated in the same direction at angular speed of ω radians per second about their common axis. Find the magnetic field (a) at a point inside the space enclosed by the inner tube r < a , and (b) at a point between the tubes b < r < c , and (c) at a point outside the tubes r > d . ( Hint: Think of copper tubes as a capacitor and find the charge density based on the voltage applied, Q = V C , C = 2 π ε 0 L ln ( c / b ) .)

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Challenge problems

The accompanying figure shows a flat, infinitely long sheet of width a that carries a current I uniformly distributed across it. Find the magnetic field at the point P, which is in the plane of the sheet and at a distance x from one edge. Test your result for the limit a 0 .

This picture shows a flat, infinitely long sheet of width a that carries a current I uniformly distributed across it. Point P is in the plane of the sheet and at a distance x from one edge.

B = μ 0 I 2 π x .

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A hypothetical current flowing in the z -direction creates the field B = C [ ( x / y 2 ) i ^ + ( 1 / y ) j ^ ] in the rectangular region of the xy -plane shown in the accompanying figure. Use Ampère’s law to find the current through the rectangle.

This figure shows the rectangular region of the xy-plane; z axis is perpendicular to the plane. Points a1 and a2 are located at the x axis. Points b1 and b2 are located at the y axis. There is an equal distance between all points.
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A nonconducting hard rubber circular disk of radius R is painted with a uniform surface charge density σ . It is rotated about its axis with angular speed ω . (a) Find the magnetic field produced at a point on the axis a distance h meters from the center of the disk. (b) Find the numerical value of magnitude of the magnetic field when σ = 1 C/m 2 , R = 20 cm , h = 2 cm , and ω = 400 rad/sec , and compare it with the magnitude of magnetic field of Earth, which is about 1/2 Gauss.

a. B = μ 0 σ ω 2 [ 2 h 2 + R 2 R 2 + h 2 −2 h ] ; b. B = 4.09 × 10 −5 T , 82% of Earth’s magnetic field

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Questions & Answers

is this book for preparing IIT or neet?
jyotirmayee Reply
is it possible to increase the temperature of a gas without adding heat to it?
gurpreet Reply
I'm not sure about it, but I think it's possible. If you add some form of energy to the system, it's a possibility. Also, if you change the pression or the volume of the system, you'll increase the kinetic energy of the system, increasing the gas temperature. I don't know if I'm correct.
playdoh
For example, if you get a syringe and close the tip(sealing the air inside), and start pumping the plunger, you'll notice that it starts getting hot. Again, I'm not sure if I am correct.
playdoh
you are right for example an adiabatic process changes all variables without external energy to yield a temperature change. (Search Otto cycle)
Brad
how to draw a diagram of a triode
Ssempala Reply
whate is fckg diagrame?
Arzoodan
why do we use integration?
Monalisa Reply
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
Heeran Reply
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
Eric Reply
what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
Electric density formula
Biren Reply
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Shikhar Reply
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
Antares Reply
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
which topic now u studying?
jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
sahoo u don't seem to understand relative physics, plz study that first.
Antares
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
Ali brother ur exactly spelling is wrong,,,,,
jyotirmayee
why the colour of tube light white?
jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
quantum mechanics is the study of the photon the light particle
Agrim
actually E=mc^2 is only the rest energy of the object and a simplified version of the expansion that covers momentum and objects close to the speed of light
Brad
quantum mechanics is the study of sub atomic particles
Brad
ofcourse
Arzoodan
all formula for calculate specific latent heat of any substance
Idowu Reply
hello
Nigar
real
Nigar
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
you both?
jyotirmayee
in which class?
jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
Bilal Reply
it is the product of electric charges and distance between the two charges
Shikhar
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
Brad
brad is separation and distance ,are they different?
Ssempala
What is actually a dipole? I know charge separated by a certain distance.... but what does that really mean? what happens in a dipole? why are the charge of same magnitude?
Monalisa
Dipoles forming as a result of the unbalanced distribution of electrons in asymmetrical molecules
Heeran
What is dielectric
Ashis Reply
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
Brad
What is the difference between specific heat capacity and heat capacity? Give the equations
elly Reply
Practice Key Terms 6

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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