Answer
Verified
428.1k+ views
Hint: The first step should be to find out the magnetic field inside the circular tube due to the current flowing along its curved surface. Then, from that value of magnetic field, we can calculate the flux linking the wire loop. With the flux, we can calculate the emf induced and then we will get the current induced in it by dividing the emf by the loop resistance. Then, by using the magnetic moment formula we will get the answer.
Formula used:
Magnetic field inside the tube,
$B=\dfrac{\mu_0 I}{L}$
Flux linked in a wire loop,
$\phi =B\pi r^2$
Emf induced,
$\varepsilon = \dfrac{d\phi}{dt}$
Magnetic moment,
$M=\pi I r^2$
Complete answer:
We have been given that the length, $L$ and radius, $R$ of the circular tube is $10m$ and $0.3m$ respectively. The current flowing through its curved surface is $I$, which varies as $I=I_0 cos(300t)$ with $I_0$ being a constant.
Now, using the Ampere’s circuital law, the magnetic field inside the tube can be written as
$B=\dfrac{\mu_0 I}{L}$,
And also, the flux linking the wire loop will be
$\phi =B\pi r^2$
Where $r$ be the radius of the wire loop placed inside the tube.
So,
$\phi = \dfrac{\mu_0 I}{L}\pi r^2=\dfrac{\mu_0 \pi r^2 I_0 cos300t}{L}$
Since, induced emf in a loop is given by,
$\varepsilon = \dfrac{d\phi}{dt}=-\dfrac{d}{dt}\left(\dfrac{\mu_0}{L}\pi r^2 I_0 cos300t\right)$
$\implies \varepsilon=\dfrac{\mu_0 \pi r^2 I_0 300sin300t}{L}$
Therefore, current induced in the loop will be,
$i=\dfrac{\varepsilon }{r'}$
Where ${r}'$ is the resistance of the loop.
$\therefore i=\dfrac{{{\mu }_{0}}\pi {{r}^{2}}{{I}_{0}}300sin300t}{Lr'}$
Since, magnetic moment of the loop is given by,
$M=\pi I r^2$
$\Rightarrow M=\dfrac{300{{\pi }^{2}}{{r}^{4}}{{\mu }_{0}}{{I}_{0}}sin300t}{Lr'}$
Now, let us substitute the given values and taking
$\pi^2 =10$
$\Rightarrow M=\dfrac{300\times 10\times (0.1)^4}{10\times 0.005}\mu_0 I_0 sin300t=6\mu_0 I_0 sin300t$
Comparing this term with the one given in the question, we get
$N=6$
So, the correct answer is “Option A”.
Note:
Initially, there is no current flowing in the wire loop. The magnetic field produced by the circular tube will induce an emf inside the loop and because of that, there will be current. The value of emf can vary with the position of the loop.
Formula used:
Magnetic field inside the tube,
$B=\dfrac{\mu_0 I}{L}$
Flux linked in a wire loop,
$\phi =B\pi r^2$
Emf induced,
$\varepsilon = \dfrac{d\phi}{dt}$
Magnetic moment,
$M=\pi I r^2$
Complete answer:
We have been given that the length, $L$ and radius, $R$ of the circular tube is $10m$ and $0.3m$ respectively. The current flowing through its curved surface is $I$, which varies as $I=I_0 cos(300t)$ with $I_0$ being a constant.
Now, using the Ampere’s circuital law, the magnetic field inside the tube can be written as
$B=\dfrac{\mu_0 I}{L}$,
And also, the flux linking the wire loop will be
$\phi =B\pi r^2$
Where $r$ be the radius of the wire loop placed inside the tube.
So,
$\phi = \dfrac{\mu_0 I}{L}\pi r^2=\dfrac{\mu_0 \pi r^2 I_0 cos300t}{L}$
Since, induced emf in a loop is given by,
$\varepsilon = \dfrac{d\phi}{dt}=-\dfrac{d}{dt}\left(\dfrac{\mu_0}{L}\pi r^2 I_0 cos300t\right)$
$\implies \varepsilon=\dfrac{\mu_0 \pi r^2 I_0 300sin300t}{L}$
Therefore, current induced in the loop will be,
$i=\dfrac{\varepsilon }{r'}$
Where ${r}'$ is the resistance of the loop.
$\therefore i=\dfrac{{{\mu }_{0}}\pi {{r}^{2}}{{I}_{0}}300sin300t}{Lr'}$
Since, magnetic moment of the loop is given by,
$M=\pi I r^2$
$\Rightarrow M=\dfrac{300{{\pi }^{2}}{{r}^{4}}{{\mu }_{0}}{{I}_{0}}sin300t}{Lr'}$
Now, let us substitute the given values and taking
$\pi^2 =10$
$\Rightarrow M=\dfrac{300\times 10\times (0.1)^4}{10\times 0.005}\mu_0 I_0 sin300t=6\mu_0 I_0 sin300t$
Comparing this term with the one given in the question, we get
$N=6$
So, the correct answer is “Option A”.
Note:
Initially, there is no current flowing in the wire loop. The magnetic field produced by the circular tube will induce an emf inside the loop and because of that, there will be current. The value of emf can vary with the position of the loop.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Advantages and disadvantages of science
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference Between Plant Cell and Animal Cell
10 examples of evaporation in daily life with explanations
Give 10 examples for herbs , shrubs , climbers , creepers
Write a letter to the principal requesting him to grant class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE