Solenoid Induction Homework: Bz & Flux Calculation

[Bryon]

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): B_z = 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

B_z = 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?

 

[Bryon]

Need the picture?

 

[Bryon]

Help?

 

[gneill]

Are you sure that the magnitude of the flux through the small loop depends upon the number of turns in that loop?

 

[Bryon]

Hmmm. I am not entirely sure. The equation for magnetic flux calls for it, but I think that is for the solenoid itself.

 

[Bryon]

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.

 

[Bryon]

I think I just answered your question! Thanks it makes a lot more sense now.