Approximating Magnetic Field for Spinning Disk with Large Distance from Center

[hitemup]

Homework Statement

I am asked to show that

$$B = \frac{\mu_0Q\omega}{2\pi R^2}[\frac{R^2+2x^2}{(R^2+x^2)^{1/2}}-2x]$$

simplifies to this

$$B \approx \frac{\mu_0}{2\pi}\frac{\mu}{x^3}$$

if x>>R

where $\mu$ is the magnetic dipole moment for a disk spinning with angular velocity $\omega$, which is

$$\mu = \frac{Q\omega R^2}{4}$$

Homework Equations

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3. The Attempt at a Solution

I ignored the R^2 in the denominator since it has become a small quantity. Then I have (R^2+2x^2)/(sqrt(x^2)) -2x
From this I get R^2/x but this equation lacks the third degree of the x.
The book has a solution for this problem as I have posted but I didn't understand it either.

 

[mfb]

I ignored the R^2 in the denominator since it has become a small quantity.

Small, but not irrelevant - you subtract two "large" numbers with a small difference from each other, so the first order of the difference is relevant. See the second line in (2), where the (inverse) denominator gets expanded up to second order (the part with "...").

 

[NascentOxygen]

It looks like they are using the first few terms of the Binomial Series to approximate (a + x)^-½

Have you come across the binomial series in your maths study? See the bottom half of this page: http://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx

 

[hitemup]

It looks like they are using the first few terms of the Binomial Series to approximate (a + x)^-½

Have you come across the binomial series in your maths study? See the bottom half of this page: http://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx

I learned it in high school and it was only for integers. This is something new but I'll try to handle it, thank you. What exactly has to be done after writing the series expansion?

 

[NascentOxygen]

You substitute the first few terms of the infinite series (writing it in the numerator) in place of the square-root expression in the denominator, then multiply and simplify.