Electromagnetic induction at the centre of a cylinder with rotating current [closed]

Question

A long circular tube of length 10 m and radius 0.3 m carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as
I=I 0 cos(300t) where I 0 is constant, If the magnetic moment of the loop is Nμ 0 I 0 sin(300t) then 'N' is

This is a problem from a past year paper of JEE advance. I tried to find the solution by integrating the induction due to small ring elements of the cylinder but it ended up being too complex.

Next I tried to search it in internet and almost everywhere it is being treated as a solenoid having 1 turn. However it does not make sense to me to treat a cylinder as a turning in solenoid and then use the formula of B=μ oNI to solve the problem (which is the formula for a long solenoid having multiple turns).

But this is the only solution I could find on the net (and similar versions of it).

My query is this:

A) Is this the correct approach if so how do we justify the aforementioned problem?

B) And if it is not correct then how to solve this problem?

Answer 1

If we assume that for a given current in the cylinder B is constant and parallel to the z axis in the inner part of the cylinder and $B=0$ outside we can make the computation for the B field way easier. This assumption doesn’t give the precise answer to the problem, but it is a good approximation, especially if Radius << Length.

Now that we have this assumption we can compute $ \oint_\gamma \vec{B(r)} \,\vec{dr} = \mu_0 \cdot I $ where $\gamma$ is a closed path which goes once through the cylinder.

Since in the middle of the cylinder $\vec{B}$ is parallel to $\vec{dr}$ and $\vec{B}=const$ we can simplify the integration to $B \cdot Lenght = \mu_0 \cdot I$ (because outside the cylinder $\vec{B}=0$)

So you can use the formula $B = \frac{\mu_0 \cdot I}{Lenght}$ which is the formula they used in the solution you linked to your post. So using these assumptions is a good approach for this problem if you solve it by hand, since otherwise it would get way too complicated.

Answer 2

After consulting some people i realised that the approach is correct. We can treat it as a solenoid whose current is I in each turn then

$$ B = \mu_0 \frac{N}{L} I $$

now the current in our solenoid is I=I0/N where I0 is the total current flowing round the cylinder.

Making this substitution we get:

B=μ0I0/L

hence it felt like this approach is taking N=1 when that really is not the case.