How Do You Calculate the Magnetic Field at a Point Along the Axis of a Solenoid?

[KracniyMyedved]

Homework Statement

A solenoid of length L and radius R lies on the y-axis between y=0 and y=L and contains N closely spaced turns carrying a steady current I. Find the magnetic field at a point along the axis as a function of distance a from the end of the solenoid.

Homework Equations

Magnetic field on the axis (distance x from center) due to a current carrying ring : $\vec{B}= \frac{\mu_o I R^2}{2(R^2 +x^2)^{3/2}}$
Biot-Savart Law: $\vec{B}=\frac{\mu_o I}{4\pi} \int_A^B \frac{\vec{ds}\mathbf{x}\vec{r}}{r^3}$

The Attempt at a Solution

My best attempt is based around modelling the solenoid as a collection of current carrying rings covering a distance of L.

$\vec{B}= \frac{\mu_o I R^2}{2(R^2 +x^2)^{3/2}}$ for each ring of current

$\vec{B}= \int_{l+a}^a \frac{\mu_o I R^2}{2(R^2 +x^2)^{3/2}}dx$

$\vec{B}= \frac{\mu_o I R^2}{2} \int_{l+a}^a \frac{dx}{(R^2 +x^2)^{3/2}}$

Integrating via trig substitution, using x = Rtanθ, which implies $dx=Rsec^2 θ dθ$ and $(R^2 + x^2)^{3/2}=R^3 sec^3 θ$

$\vec{B}= \frac{\mu_o I R^2}{2} \int_{x=l+a}^{x=a} \frac{Rsec^2 \theta d\theta}{R^3 sec^3 \theta}$

$\vec{B}= \frac{\mu_o I}{2} \int_{x=l+a}^{x=a} \frac{d\theta}{sec\theta}$

$\vec{B}= \frac{\mu_o I}{2} \int_{x=l+a}^{x=a} cos\theta d\theta$

$\vec{B}= \frac{\mu_o I}{2} sin\theta$ for x from x=l+a to x=a

Constructing a triangle from the substitution let's us find $sin\theta$ in terms of R and x:
$sin\theta = \frac{x}{(x^2 + R^2)^{1/2}}$

Finally,

$\vec{B}= \frac{\mu_o I}{2}(\frac{a}{(a^2+R^2)^{1/2}} - \frac{l+a}{((l+a)^2 +R^2)^{1/2}})$

I'm reasonably sure this is wrong because the next part of the question implies a dependence on N, as well I think my modelling the solenoid as I did was a bit shaky, but I cannot think of another way to go about this.

 

[Simon Bridge]

I'm reasonably sure this is wrong because the next part of the question implies a dependence on N, as well I think my modelling the solenoid as I did was a bit shaky, but I cannot think of another way to go about this.

Well you should expect a dependence on N wouldn;t you? After all, the field due to 2 loops must be different from the field due to 1 loop right?

This is a standard problem - so you could just look it up and see how other people approach it.
i.e. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

 

[PhysicoRaj]

My best attempt is based around modelling the solenoid as a collection of current carrying rings covering a distance of L.
$\vec{B}= \frac{\mu_o I R^2}{2(R^2 +x^2)^{3/2}}$ for each ring of current

$\vec{B}= \int_{l+a}^a \frac{\mu_o I R^2}{2(R^2 +x^2)^{3/2}}dx$

What you are actually doing here is integrating the field due to small section of the solenoid of length dx (which you later substitute the limits for the length of the solenoid) which is not necessarily a single loop, but consists of ndx loops, n being the loops per unit length.