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
Alarge plate is fabricated from a steel alloy that has a plane strain fracture toughness of 82.4 MPa m1/2. If the plate is expos
Korvikt [17]

Answer:

minimum length of a surface crack is 18.3 mm

Explanation:

Given data

plane strain fracture toughness K = 82.4 MPa m1/2

stress σ = 345 MPa

Y = 1

to find out

the minimum length of a surface crack

solution

we will calculate length by this formula

length = 1/π ( K / σ Y)²

put all value

length = 1/π ( K / σ Y)²

length = 1/π ( 82.4 10^{3/2} / 345× 1)²

length = 18.3 mm

minimum length of a surface crack is 18.3 mm

4 0
3 years ago
An airplane travels 3,260 kilometers in 4 hours. What is the airplane’s average speed?
OLga [1]
3260÷4=815 which is you average seed
4 0
3 years ago
If a Stone with an of original velocity of zero is falling from a ledge and takes eight seconds to hit the ground what is the fi
DaniilM [7]
Let t=time to reach the ground=8 secs, g= acceleration of gravity. The speed v on reaching the ground is gt=8g=78.4 m/s where g=9.8 m/s/s approx.
3 0
3 years ago
The electron has the positron as its antiparticle.<br><br> A True<br> B False
Sphinxa [80]
The answer would be true
8 0
3 years ago
Read 2 more answers
Which statement best summarizes Gregor Mendel’s contribution to science?
Rama09 [41]
Hey there!

<span>Which statement best summarizes Gregor Mendel’s contribution to science?

Mendel is considered the "father of genetics". He did NOT increse the number of peas </span>
<span>produced by a single plant using cross-breeding techniques, discover grean peas and yellow peas, or invent the green pea by cross-breeding thousands of pea plants.

This means your answer is option D.

Hope this helps
Have a great day (:
</span>
8 0
3 years ago
Read 2 more answers
Other questions:
  • The diagram shows how air moves near the coast on a warm day. Explain why air moves as shown on the diagram. 5 marks
    13·1 answer
  • Which scenario requires the most power?
    14·1 answer
  • How to draw the ray diagram? Pls answer
    10·1 answer
  • A bumblebee is flying to the right when a breeze causes the bee to slow down with a constant leftward acceleration of magnitude
    13·1 answer
  • A car starts from rest at a stop sign. It accelerates at 3.8 m/s 2 m/s2 for 6.0 s, coasts for 1.6 ss , and then slows down at a
    14·1 answer
  • Column A is in the x-axis, and column B is on the y-axis. Which titles should replace A and B
    9·1 answer
  • This pendulum is at<br><br> minimum displacement<br> maximum displacement<br> equilibrium
    9·2 answers
  • How much momentum does Elena's 1200kg car have if it is traveling with a velocity of 15 m/s N?
    5·1 answer
  • A boy holds a toy soldier in from of a concave mirror. The focal length of the mirror is 0.45 m and the boy holds the toy soldie
    6·1 answer
  • As a person pushes a box across a floor, the energy from the person's moving arm is transferred to the box, and the box and the
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!