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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

grin007 [14]

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,

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

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}
$

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)=
\\\\2\pi\,B_{in} l=2\pi\mu_0k \frac{r^3}{3}
\\\\B_{in}=\frac{\mu_0kl^2}{3}
$

now using equation 1, putting the value of k,

$B_{in} = \frac{\mu_{0} l^2 }{3 } \,\,\, \frac{3I}{2 \pi a^3}
\\\\B_{in} = \frac{ \mu_{0} I l^2}{2 \pi a^3}
$

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}
$

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)
\\\\2\pi\,B_{out}r=2\pi\mu_0k\frac{a^3}{3}
\\\\B_{out}=\frac{\mu_0ka^3}{3r}
$

again using,equaiton 1,

$B_{out}= \mu_0 \frac{a^3}{3r} \times \frac{3 I}{2 \pi a^3}
\\\\B_{out} = \frac{ \mu_{0} I}{2 \pi r}$

8 0

3 years ago

Wolfgang pauli hypothesized an exclusion principle. This principle says two electrons in an atom cannot have the same what?.

tatiyna

No two electrons in an atom or molecule may have the same four electronic quantum numbers, according to the Pauli Exclusion Principle. Only two electrons can fit into an orbital at a time, hence they must have opposing spins.

<h3>What is Pauli's exclusion principle ?</h3>

According to Pauli's Exclusion Principle, no two electrons in the same atom can have values for all four of their quantum numbers that are exactly the same. In other words, two electrons in the same orbital must have opposing spins and no more than two electrons can occupy the same orbital.

The reason it is known as the exclusion principle is because it states that all other electrons in an atom are excluded if one electron in the atom has the same specific values for all four quantum numbers.

Learn more about Pauli's exclusion principle here:

brainly.com/question/2623936

#SPJ4

3 0

1 year ago

What happens to the ball's velocity while the ball is traveling upwards?

Bess [88]

If the ball does not have a propeller or jet engine on it, then it is an object
in free fall. That means its downward speed grows by 9.8 m/s for every
second that it's in the air.

If it happens to be traveling upward at the moment, then that won't last long.
Its upward speed is decreasing by 9.8 m/s every second. It will eventually
run out of upward gas and start moving downward. At that instant, you might
say that the direction of its velocity has changed by 180 degrees.

7 0

3 years ago

Convert 463.833 m/s to km/ min

Alexus [3.1K]

Answer:

27.82998 km/min

Explanation:

To convert m/sec into km/hr, multiply the number by 18 and then divide it by 5.

8 0

3 years ago

the hot gas lines need to be insulated? A never B always C if they are exposed to cold temperatures that could cause the refrige

muminat

The answer is a because it never needs to be insulated

8 0

3 years ago