Solenoid Induction Homework: Bz & Flux Calculation

In summary, the solenoid is producing a magnetic field that is varying with time according to the expression: I1(t) = 0.45 A + 0.14 (A/s) t. The magnetic flux through the loop at t = 3 s is 0.87A.
  • #1
Bryon
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Homework Statement


A solenoid has length L = 20 cm, radius 6 cm, and N1 = 4500 turns; its axis coincides with the z-axis. A circular conducting loop containing N2 = 11 turns of radius a = 1 cm is centered inside the solenoid; the plane of the loop makes a 30° angle with respect to the z-axis.

The current in the windings of the solenoid is varying with time according to the expression
I1(t) = 0.45 A + 0.14 (A/s) t.


Homework Equations



Biot-Savart(Center of a solenoid): Bz = u0nI
Magnetic flux: ϕ = ∫BdA= NBAcosϑ

The Attempt at a Solution



(a) Calculate the magnitude of the magnetic flux Φm through the loop at t = 3 s. (Absolute value)

I1(3s) = 0.45 A + 0.14 (A/s) (3s) = 0.87A

Bz = 4*pi*10^(-7)*(4500/0.2)*0.87 = 0.02459867 T

ϕ = (0.02459867 )*pi*(.01^2)*11*cos(30) = 7.3618137e-5 Tm^2

I did something wrong and I am not quite sure where. Any ideas?

I think i understand the concept. Since there is a small loop inside the solenoid at angle ϑ, I need to find the magnetic field at the center due to the current in the solenoid. Then I just have to find the total flux through the total area of the small circular loop that is sitting in the middle of the solenoid.

The question does provide a hint: Approximate this finite solenoid by an equivalent infinite solenoid to find the magnetic field in the central region.

Does this mean I can use a portion of the number of turns over a small distance to find the magnetic field?
 
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  • #2
Need the picture?
 
  • #3
Help?
 
  • #4
Are you sure that the magnitude of the flux through the small loop depends upon the number of turns in that loop?
 
  • #5
Hmmm. I am not entirely sure. The equation for magnetic flux calls for it, but I think that is for the solenoid itself.
 
  • #6
I know that the magnetic field is uniform with in the solenoid near the center, and that would mean that the loop will see what ever its area is.
 
  • #7
I think I just answered your question! Thanks it makes a lot more sense now.
 

Related to Solenoid Induction Homework: Bz & Flux Calculation

1. What is the purpose of calculating Bz and flux in solenoid induction?

The purpose of calculating Bz and flux in solenoid induction is to understand and predict the behavior of electromagnetic fields in a solenoid. Bz refers to the magnetic field strength along the axis of the solenoid, while flux refers to the total amount of magnetic field passing through a given surface. These calculations allow scientists to analyze and optimize the performance of solenoids for various applications, such as in motors, generators, and other electrical devices.

2. How is Bz calculated in solenoid induction?

Bz is calculated using the formula Bz = μ0 * n * I, where μ0 is the permeability of free space, n is the number of turns in the solenoid, and I is the current flowing through the solenoid. This formula takes into account the physical properties of the solenoid, such as its length and number of turns, as well as the amount of current passing through it, to determine the strength of the magnetic field along its axis.

3. What factors affect the calculation of flux in solenoid induction?

The calculation of flux in solenoid induction is affected by several factors, including the number of turns in the solenoid, the strength of the magnetic field, and the surface area of the surface through which the flux is passing. Additionally, the orientation and shape of the solenoid can also impact the calculation of flux.

4. How does solenoid material affect Bz and flux calculations?

The type of material used in a solenoid can affect the Bz and flux calculations. Certain materials, such as iron, have a higher permeability than air, which means they can increase the strength of the magnetic field and the amount of flux passing through the solenoid. However, the material can also impact the resistance and other properties of the solenoid, which can affect the overall performance and efficiency of the device.

5. What are some real-world applications of Bz and flux calculations in solenoid induction?

Bz and flux calculations in solenoid induction have numerous real-world applications, including in the design and optimization of motors, generators, and other electrical devices. They are also used in the study of magnetic fields in various materials and industries, such as in the production of magnetic tapes and hard drives. Additionally, these calculations are crucial in the development of medical imaging techniques, such as magnetic resonance imaging (MRI).

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