Binomial expansion of relativistic formula

[seboastien]

Homework Statement

The speed v of electrons from a high energy accelerator is very near the speed of light c. Given the voltage V of the accelerator, calculate the ratio v/c. The relativistic formulafor this calculation is (see relevant equations)

Use the binomial series to find (1-v/c) in terms of V

Homework Equations

v/c = (1-(1/4V^2))^0.5

The Attempt at a Solution

v/c = (1-(4v^2)^-1)^0.5 = (approx) [1- (1/8V^2) + (3/128V^4) - (15/3072V^6)]

therefore;

1-v/c= (1/8V^2) - (3/128V^4) + (15/3072V^6) = 1.25x10^(-17)

 

[mighty2000]

V^2 - 2.34x10^(-20)V^4 + 4.88x10^(-24)V^6

Thank you for your post! Your approach using the binomial series is correct. However, there are a few minor errors in your final expression for (1-v/c). The correct expression is:

1-v/c = (1/8V^2) - (3/128V^4) + (5/1024V^6) - (35/32768V^8) + ...

You can see that the pattern is that the numerator alternates between 1 and -1, and the denominator increases by a power of 2 each term. Also, the first term in your expression should be (1/8V^2) instead of (1/4V^2).

Overall, your work shows a good understanding of the concept and a good use of the binomial series. Keep up the good work!