How does one go about integrating something like $$V = \int_{a}^{b}\frac{Q}{2\pi r\epsilon_{0}\epsilon_{r}}dr$$Where the values of $a$,$b$,$\epsilon$,$Q$ are given and V is the potential difference
Thanks in advance.
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We simply use standard calculus techniques. Remember, $\varepsilon_{0}$, $\varepsilon_{r}$, $2\pi$ and $Q$ are real-valued constants. So we can rewrite the integral: $$V=\int_{a}^{b}{\frac{Q}{2\pi r\varepsilon_{0}\varepsilon_{r}}\:dr}=\frac{Q}{2\pi\varepsilon_{0}\varepsilon_{r}}\int_{a}^{b}{\frac{1}{r}\:dr}$$ We also know that $\int{\frac{1}{r}\:dr}=\ln{|r|}+c_{1}$, so we have: $$V=\frac{Q}{2\pi\varepsilon_{0}\varepsilon_{r}}\left(\ln|b|-\ln|a|\right)=\frac{Q}{2\pi\varepsilon_{0}\varepsilon_{r}}\ln{\left|\frac{b}{a}\right|}$$ |
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