Magnetic Field in Solenoid question

[Oshada]

Homework Statement

Your physics lab demonstrator hands you a spool of copper wire and a battery (voltage V) and asks you to wind the wire around a hollow, cylindrical, cardboard tube (radius a) to make a solenoid of length 2L.

a) What is the amplitude of the magnetic field at the centre of the carboard tube (ie., on the axis, halfway between the ends) due to a single turn of wire at a distance z from the centre?

b) What is the amplitude of the magnetic field at the centre of the solenoid of length 2L?

c) How long should the solenoid be so that the magnetic field in PART B equals 60% of the value in an infinite solenoid?

d) In real life, for L much bigger than a, how does the magnetic field in PART B scale with L?

e) Suppose your physics lab class lasts for a very long time - long enough for you to wind a semi-infinite solenoid. What is the on-axis magnetic field at the end of the solenoid closest to you (i.e. not at infinity)?

f) What is the on-axis magnetic field a distance 2a beyond the end of the semi-infinite solenoid, as a fraction of your answer to PART E?

Homework Equations

Biot-Savart Law, Ampere's Law (maybe some others?)

The Attempt at a Solution

I can do a) using Biot-Savart Law, and b) using Ampere's Law. However I'm confused by the rest of the sections. Any help is welcome!

 

[Oshada]

Bump. Anyone?

 

[thebigman]

what did you get for a) and b)? did you just use B = μ0nI ?

 

[thebigman]

http://www.cramster.com/profile-7116314
that should help

 

[rwisz]

I would like to first commend you for your understanding and application of the Biot-Savart Law and Ampere's Law. These are indeed the relevant equations to use in solving this problem. To answer the additional questions, we can use some other relevant equations and concepts from electromagnetism.

c) To find the length of the solenoid for the magnetic field to be 60% of the value in an infinite solenoid, we can use the equation B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. Since we are given the voltage and the radius of the solenoid, we can use Ohm's Law to find the current. Then, we can use the given information about the magnetic field to solve for the number of turns per unit length, which can be used to find the length of the solenoid.

d) In real life, as the length of the solenoid (L) becomes much larger than the radius (a), the magnetic field at the centre of the solenoid will approach the value of an infinite solenoid, which is given by B = μ₀nI. This is because as L increases, the solenoid becomes more closely packed and the magnetic field lines become more parallel, leading to a more uniform and stronger magnetic field.

e) To find the on-axis magnetic field at the end of the semi-infinite solenoid closest to you, we can use the equation B = μ₀nI. However, since the solenoid is semi-infinite, we need to consider the contribution of all the turns of wire before the end of the solenoid. This can be done by integrating the Biot-Savart Law over the entire length of the solenoid. This will give us the total magnetic field at the end of the solenoid, and we can then use the given information to find the on-axis magnetic field at the end closest to you.

f) To find the on-axis magnetic field a distance 2a beyond the end of the semi-infinite solenoid, we can use the same method as in part e), but with the distance z being 2a instead of 0. This will give us the total magnetic field at a distance 2a beyond the end of the solenoid