# How does one calculate the flux density B at end of solenoid?

• Nile Anderson
In summary, the flux density at the end of a solenoid with 12 turns per cm and a current of 3A is 144π×10-5.
Nile Anderson

## Homework Statement

I came across a recent problem that asked me to calculate the flux density at the end of a solenoid. I was given the current 3 A , the number of turns per unit length , 12 cm-1 and the using the permeability of free space as 4π × 10^-7

## Homework Equations

The equation I used for this is B=μnI , where n is the number of turns per unit length ( in per meter) and I current.

## The Attempt at a Solution

I got 144π × 10^-5, this however was not an option and so I was wondering if there were some other factor I had to account for as it is the end of the coil and not for the midpoint for which the equation is defined.

You need to show your working so helpers can first discount that you may have made a simple mistake in the mathematics.

NascentOxygen said:
You need to show your working so helpers can first discount that you may have made a simple mistake in the mathematics.
With all due respect sir, I have no problems using a calculate but ok , never the less:
You need to convert the 12 per cm to per m which gives you 12/10^-2=1200
B= 4π×10-7 ×1200 × 3 = 144π×10-5

Nile Anderson said:
I got 144π × 10^-5, this however was not an option and so I was wondering if there were some other factor I had to account for as it is the end of the coil and not for the midpoint for which the equation is defined.
Presumably this is a multiple choice style of question?

Can you quote the problem exactly along with the selection of answers?

It sometimes happens that a problem is "refreshed" with a slight change in values in order to make it "new" (so last year's answers can't be copied). Occasionally the "refresh" is imperfect: either the new answer is not inserted in the list of answers or the person making the change made an error in calculation...

Nile Anderson
B=µnI gives the flux density midway between the two ends of the solenoid.

Follow the link in this thread to obtain the formula for B near the end of a solenoid. If I read it correctly, you'll find that right at the end they'll differ by a factor of approx. 2

Nile Anderson
The question said "What is the magnetic flux density near the end of a solenoid that has 12 turns per centimeter and carrying a current of 3A? (Assume μo =4π×10-7)
A) 144π×10-7) T
B) 1.44π×10-7) T
C) 144π μT
D) 144π T
gneill said:
Presumably this is a multiple choice style of question?

Can you quote the problem exactly along with the selection of answers?

It sometimes happens that a problem is "refreshed" with a slight change in values in order to make it "new" (so last year's answers can't be copied). Occasionally the "refresh" is imperfect: either the new answer is not inserted in the list of answers or the person making the change made an error in calculation...

Nile Anderson said:
A) 144π×10-7) T
B) 1.44π×10-7) T
C) 144π μT
D) 144π T

Looks like correct answer is: (E) none of the above

NascentOxygen said:
Looks like correct answer is: (E) none of the above
Lol yes , a reasonable conclusion

The two factors are limits for very long and very short solenoid relative to ring radius, so I think that you ask for the assumptions to use on your class level.

Nile Anderson said:
The question said "What is the magnetic flux density near the end of a solenoid that has 12 turns per centimeter and carrying a current of 3A? (Assume μo =4π×10-7)
A) 144π×10-7) T
B) 1.44π×10-7) T
C) 144π μT
D) 144π T
Length of solenoid? 1cm? 1m?

If length << radius, B = μ0IR/2
which is of no help since radius R is not given.
What you do is sum the B fields due to each wire turn using Biot-Savart. This is most readily done by the appropriate integral, so what you're doing here is assuming a differentially small wire spacing then integrating over distance from the solenoid's end (x=0) to x=infinity, and pre-multiplying by 1/(actual wire spacing).

Last edited:
Imagine two exacly identical coils carrying the same current in the same direction. B along the axis of each coil is given by by Nile in post one. Let B at the end points of the coils have the value Be. Now imagine that the two coils are pushed together end to end so as to make a longer coil. B along the axis will have the same value as before. Now consider the midpoint of the axis of this coil, where two ends of the smaller coils meet. B at this point can be considered to be the sum of B at the two ends of the smaller coils and therefore B = 2Be. In other words B at the end points of a long coil is half the value of B within the coil.

## 1. What is flux density?

Flux density, also known as magnetic flux density or magnetic induction, is a measure of the strength of a magnetic field. It is represented by the symbol B and is measured in units of tesla (T) in the SI system.

## 2. How is flux density calculated?

The flux density at the end of a solenoid can be calculated using the equation B = μ₀nI, where μ₀ is the permeability of free space (4π x 10⁻⁷ T∙m/A), n is the number of turns per unit length of the solenoid, and I is the current flowing through the solenoid. This equation assumes that the solenoid is infinitely long and has a uniform magnetic field at its center.

## 3. What factors affect the flux density at the end of a solenoid?

The flux density at the end of a solenoid is affected by the number of turns per unit length (n), the current flowing through the solenoid (I), and the permeability of free space (μ₀). Additionally, the length and radius of the solenoid also play a role in determining the flux density.

## 4. How does the flux density change with distance from the end of the solenoid?

The flux density at the end of the solenoid is the strongest, and it decreases as you move away from the end. This is because the magnetic field lines spread out as they move away from the solenoid, resulting in a decrease in flux density.

## 5. What are some real-world applications of calculating flux density at the end of a solenoid?

Flux density calculations are important in many technological applications, such as in the design of magnetic sensors and motors. They are also used in medical imaging techniques like magnetic resonance imaging (MRI) and in particle accelerators in physics experiments.

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