Physics Problems & Solutions: Electromagnetism

If n is an integer ranging from 1 to infinity, is an angular frequency, and t is time, then the Fourier series for a square wave, as shown above, is given by which of the following?

A. $V(t)=\frac{4}{\pi}\sum_{1}^{\infty}\frac{1}{n}\sin(n\omega t)$

B. $V(t)=\frac{4}{\pi}\sum_{0}^{\infty}\frac{1}{(2n+1)}\sin(\left(2n+1\right)\omega t)$

C. $V(t)=\frac{4}{\pi}\sum_{1}^{\infty}\frac{1}{n}\cos(n\omega t)$

D. $V(t)=\frac{4}{\pi}\sum_{0}^{\infty}\frac{1}{(2n+1)}\cos(\left(2n+1\right)\omega t)$

E. $V(t)=-\frac{4}{\pi}+\frac{4}{\pi}\sum_{1}^{\infty}\frac{1}{n^2}\cos(n\omega t)$

(GR9277 #39)

Solution:

The sine function is an odd function.
The cosine function is an even function.
The function in this problem is an odd square wave function.
→ C, D, E are FALSE.

Both A and B are the same, thus the right solutions. However, A is more generalized and B is the simplified answer.

For Answer A, the sine term vanishes when n is even (n = 0, 2, 4, ...) → rewrite n for odd term as 2n +1 with n starts at 0 → the answer can be simplified to B.

Answer: B

Physics Problems & Solutions

Electromagnetism - Fourier Series

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