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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

four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
Aniket Reply
the study of heat an other form of energy.
John
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
ing
It is abranch of physical chemistry which deals with the interconversion of all form of energy
Vishal
what is colamb,s law.?
Muhammad Reply
it is a low studied the force between 2 charges F=q.q`\r.r
Mostafa
what is the formula of del in cylindrical, polar media
Birengeso Reply
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.
MAXWELL Reply
what is flux
Bundi Reply
Total number of field lines crossing the surface area
Kamru
Basically flux in general is amount of anything...In Electricity and Magnetism it is the total no..of electric field lines or Magnetic field lines passing normally through the suface
prince
what is temperature change
Celine
a bottle of soft drink was removed from refrigerator and after some time, it was observed that its temperature has increased by 15 degree Celsius, what is the temperature change in degree Fahrenheit and degree Celsius
Celine
process whereby the degree of hotness of a body (or medium) changes
Salim
Q=mcΔT
Salim
where The letter "Q" is the heat transferred in an exchange in calories, "m" is the mass of the substance being heated in grams, "c" is its specific heat capacity and the static value, and "ΔT" is its change in temperature in degrees Celsius to reflect the change in temperature.
Salim
what was the temperature of the soft drink when it was removed ?
Salim
15 degree Celsius
Celine
15 degree
Celine
ok I think is just conversion
Salim
15 degree Celsius to Fahrenheit
Salim
0 degree Celsius = 32 Fahrenheit
Salim
15 degree Celsius = (15×1.8)+32 =59 Fahrenheit
Salim
I dont understand
Celine
the question said you should convert 15 degree Celsius to Fahrenheit
Salim
To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.
Salim
what is d final ans for Fahrenheit and Celsius
Celine
it said what is temperature change in Fahrenheit and Celsius
Celine
the 15 is already in Celsius
Salim
So the final answer for Fahrenheit is 59
Salim
what is d final ans for Fahrenheit and Celsius
Celine
what are the effects of placing a dielectric between the plates of a capacitor
Bundi Reply
increase the capacitance.
Jorge
besides increasing the capacitance, is there any?
Bundi
mechanical stiffness and small size
Jorge
so as to increase the capacitance of a capacitor
Rahma
why for an ideal gas internal energy is directly proportional to thermodynamics temperature?
Anne Reply
two charged particles are 8.45cm apart. They are moved and the force on each of them is found to have tripled. How far are they now?
Martin Reply
what is flux
Bundi
Bundi, flux is the number of electric field crossing a surface area
Mubanga
you right
Muhammad
martin,F/F`=(r`×r`)÷(r×r)
Mostafa
determining dimensional correctness
PATRICK Reply
determine dimensional correctness of,T=2π√L/g
PATRICK
somebody help me answer the question above
PATRICK
d=dQ+w
Muhammad
calculate the heat flow per square meter through a mineral roll insulation 5cm thick if the temperature on the two surfaces are 30degree Celsius and 20 degree Celsius respectively. thermal conduction of mineral roll is 0.04
akuribire Reply
what are the elementary compositions of a cell?
jackson Reply
poles, chemical
prabir
when a current pass through a material does the velocity varies
lovet Reply
no.
prabir
what is spin entropy ?and disorder in ferromagnetic material
Nepal 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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