What Is the Induced EMF in the Coil?

[Ling2]

A solenoid of length 45 cm has 340 turns of radius 2.2 cm. A tightly wound coil with 16 turns of radius 4.4 cm is at the center of the solenoid. The axes of the coil and solenoid coincide. Find the emf induced in the coil if the current in the solenoid varies according to I=4.6sin(50∏t)A.

Answer: __________cos(50∏t) mV

Comments:
I can't seem to find the correct answer for the magnetic field of the solenoid.
The formula I use is:
B=0.5μ₀nI(sinθ₂-sinθ₁)
The area to use for all equations is the area of the solenoid since it is the area of magnetic field lines felt by the coil.
The flux, denoted ∅ is: ∅=BAcosσ and σ=0^o, therefore ∅=BA
and the EMF=-N(d∅/dt)=-NA(dB/dt)

I really appreciate any help, thank you!

 

[grzz]

B on the axis and at the centre of a 'long' solenoid is given by B = $\mu$$_{o}$nI where n = number of turns/length of solenoid.

 

[Ling2]

Assuming the solenoid is long an using that formula, my answer is wrong. Is there anything missing to the comments I added for solving this problem?

 

[grzz]

Can you show the working for dB/dt?

 

[paranormal]

I understand your frustration with finding the correct solution for the magnetic field of the solenoid. However, it is important to note that there are different formulas and methods for calculating the magnetic field in a solenoid, and the one you are using may not be the most suitable for this specific problem.

In this case, I would recommend using the formula for the magnetic field inside a solenoid, which is B=μ0nI. In this formula, μ0 is the permeability of free space, n is the number of turns per unit length (in this case, 340/0.45 m), and I is the current in the solenoid.

Using this formula, we can calculate the magnetic field at the center of the solenoid, where the tightly wound coil is located. Plugging in the values, we get B= (4π×10^-7)(340/0.45)(4.6sin(50πt)) = 0.033sin(50πt) T.

To find the EMF induced in the coil, we can use Faraday's Law of Induction, which states that the EMF induced in a coil is equal to the rate of change of magnetic flux through the coil. In this case, the coil is at the center of the solenoid, so we can use the same magnetic field value as before. Therefore, the EMF induced in the coil is given by EMF = -N(d∅/dt) = -N(A(dB/dt)) = -16(0.033cos(50πt))(50π) = -2.64cos(50πt) mV.

I hope this helps clarify the solution for this problem. It is always important to carefully consider which formula and method is most suitable for a given problem, as different situations may require different approaches. Keep up the good work in your scientific endeavors!