1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
PSYCHO15rus [73]
3 years ago
13

A steady current I flows through a wire of radius a. The current density in the wire varies with r as J = kr, where k is a const

ant and r is the distance from the axis of the wire. Find expressions for the magnitudes of the magnetic field inside and outside the wire as a function of r. (Hint: Find the current through an Ampèrian loop of radius r using Ithru = J · dA. Use the following as necessary: μ0, I, a and r. Do not substitute numerical values; use variables only.)a. B inside=?b. B outside=?
Physics
1 answer:
grin007 [14]3 years ago
8 0

Answer:

Explanation:

we can consider an element of radius r < a and thickness dr.  and Area of this element is

dA=2\pi r dr

since current density is given

J=kr

then , current through this element will be,

di_{thru}=JdA=(kr)(2\pi\,r\,dr)=2\pi\,kr^2\,dr

integrating on both sides between the appropriate limits,

\int_0^Idi_{thru}=\int_0^a2\pi\,kr^2\,dr&#10;\\\\&#10;I=\frac{2\pi\,ka^3}{3} -------------------------------(1)

Magnetic field can be found by using Ampere's law

\oint{\vec{B}\cdot\,d\vec{l}}=\mu_0\,i_{enc}

for points inside the wire ( r<a)

now, consider a point at a distance 'r' from the center of wire. The appropriate Amperian loop is a circle of radius r.

by applying the Ampere's law, we can write

\oint{\vec{B}_{in}\cdot\,d\vec{l}}=\mu_0\,i_{enc}&#10;

by symmetry \vec{B} will be of uniform magnitude on this loop and it's direction will be tangential to the loop.

Hence,

B_{in}\times2\pi\,l=\mu_0\int_0^r(kr)(2\pi\,r\,dr)=&#10;\\\\2\pi\,B_{in} l=2\pi\mu_0k \frac{r^3}{3}&#10;\\\\B_{in}=\frac{\mu_0kl^2}{3}&#10;

now using equation 1, putting the value of k,

B_{in} = \frac{\mu_{0} l^2 }{3 } \,\,\, \frac{3I}{2 \pi a^3}&#10;\\\\B_{in} = \frac{ \mu_{0} I l^2}{2 \pi a^3}&#10;

B)

now, for points outside the wire ( r>a)

consider a point at a distance 'r' from the center of wire. The appropriate Amperian loop is a circle of radius l.

applying the Ampere's law

\oint{\vec{B}_{out}\cdot\,d\vec{l}}=\mu_0\,i_{enc}&#10;

by symmetry \vec{B} will be of uniform magnitude on this loop and it's direction will be tangential to the loop. Hence

B_{out}\times2\pi\,r=\mu_0\int_0^a(kr)(2\pi\,r\,dr)&#10;\\\\2\pi\,B_{out}r=2\pi\mu_0k\frac{a^3}{3}&#10;\\\\B_{out}=\frac{\mu_0ka^3}{3r}&#10;

again using,equaiton 1,

B_{out}= \mu_0 \frac{a^3}{3r} \times \frac{3 I}{2 \pi a^3}&#10;\\\\B_{out} = \frac{ \mu_{0} I}{2 \pi r}

You might be interested in
Two tugboats are moving a barge. Tugboat A pushes on the barge with a force of 3000n. Tugboat B pulls with a force of 5000 Newto
kenny6666 [7]

The net force on the barge is 8000 N

Explanation:

In order to find the net force on the badge, we have to use the rules of vector addition, since force is a vector quantity.

In this problem, we have two forces:

  • The force of tugboat A, F_A = 3000 N, acting in a certain direction
  • The force of tugboat B, F_B = 5000 N, also acting in the same direction

Since the two forces act in the same direction, this means that we can simply add their magnitudes to find the net combined force on the barge. Therefore, we get

F=F_A+F_B = 3000 + 5000 = 8000 N

and the direction is the same as the direction of the two forces.

Learn more about forces:

brainly.com/question/11179347

brainly.com/question/6268248

#LearnwithBrainly

5 0
3 years ago
How did the continental Shelf form? Please help, thank you!! :)
shutvik [7]
The movements of the tectonic plates
6 0
3 years ago
Thalia is drafting a plan to move a large, perfect sphere concrete sculpture that is in front of her office building. Describe t
Marina86 [1]
For the answer to this question, 
Thalia must consider the weight of the object and the mass of the sculpture. Weight and mass are different things. She should also consider the time on how long it will take to move it and where she'll move it.
4 0
3 years ago
Read 2 more answers
Running up a hill instead of a flat surface will increase which principle of fitness?
musickatia [10]

Answer:

a

Explanation:

5 0
3 years ago
Read 2 more answers
Me podrían dar la respuesta y el proceso?
Nat2105 [25]

Answer:

I don’t understand Espanol

Explanation:

sorry

7 0
3 years ago
Other questions:
  • You are exploring a distant planet. When your spaceship is in a circular orbit at a distance of 630 km above the planet's surfac
    11·1 answer
  • A magnetic pickup is used to detect the shaft speed of an experimental high speed compressor. a gear with 12 teeth is used to ex
    9·2 answers
  • Which property of light is not explained by the wave model of light?
    12·1 answer
  • What is the speed of a wave with a frequency of 2 Hz and a wavelength of 87 m?
    7·2 answers
  • A mass M suspended by a spring with force constant k has aperiod T when set into oscillations on Earth. Its period on Mars,whose
    10·1 answer
  • What do we mean by "Ohmic"?
    6·1 answer
  • What did Rutherford’s model of the atom include that Thomson’s model did not have?
    15·2 answers
  • How long will it take Matthew to jog 321 m at 5.8m/s?
    11·1 answer
  • Describe What is the shape of a velocity time graph for an object with constant acceleration
    5·1 answer
  • A grindstone increases in angular speed from 4.00 rad/s to 12.00rad/s in 4.00 s. Through what angle does it turn during that tim
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!