0

The task goes as following: Calculate the amount of charge $Q$ of a unevenly charged semicircular ring with radiuses $a$ and $b$ if $\eta=\eta_0\sin\phi, 0<\phi<\pi$.

I started off by calculating the infinitesimally small surface $dS$ charged with infinitesimally small charge $dQ$.In this case that surface would be difference between two circular snippets.

$dS=\frac{b^2\cdot\pi\cdot d\phi}{360°} - \frac{a^2\cdot\pi\cdot d\phi}{360°}$

$dS=(b^2-a^2)\frac{\pi\cdot d \phi}{360°}$

and the very next step in the textbook solution is: $dS=(b^2-a^2)\frac{d \phi}{2}$

How did $\pi$ switch from being $3.14$ (coming from $S_l=\frac{r^2\pi \cdot \alpha}{360°}$) to $180°$ (because $\frac{\pi}{360°}$ became $\frac{1}{2}$ in the next step)?

After this I continue with the integration $Q=\int_0^{\pi}\eta_0 \cdot \sin \phi \cdot (b^2-a^2) \frac{d\phi}{2})$ and so on, but the step mentioned above is not clear to me at all.

Qmechanic
  • 201,751
0lt
  • 123
  • 5
    It's very poor style to express angles in degrees in physical calculations. 360 degrees is $2\pi$, 180 is $\pi$ and 90 is $\pi/2$. You have to get used to that and stop using degrees. If you need them for a technical application, you convert them at the beginning and at the end of the calculation, but everywhere between you should stick to one system. – CuriousOne Oct 31 '15 at 22:08
  • The Wikipedia page for "degree" explains the conversion of radians to degree and vice versa. – ACuriousMind Oct 31 '15 at 22:10
  • In most cases it makes sense to simply consider the symbol ° to represent the number π/180. – kasperd Oct 31 '15 at 22:20
  • @kasperd Thank you. I was confused about $\pi$ meaning both radians and length in the same expression. I am familiar with conversion but this was somewhat odd to me. Thanks again. – 0lt Oct 31 '15 at 22:23
  • 1
    A degree is defined to be $\pi/180$. – WillO Nov 01 '15 at 00:05

1 Answers1

1

You can see the degree symbol ° as a multiplicative factor being $\pi/180$, a little bit like the % symbol is a multiplicative factor being $1/100$. I can transform $12.3\%$ into $0.123$ because $12.3\%=12.3*1/100=12.3/100=0.123$. Similarily, I can transform $\pi/360°$ into $1/2$ because $\pi/360°=\pi/(360\pi/180)=180\pi/360\pi=180/360=1/2$.