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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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    I should have written $\int_{a}^{b}\frac{1}{r}dr=\ln \left\vert b\right\vert -\ln \left\vert a\right\vert $2012-08-02

1 Answers 1

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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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    Given that we can presume $r$ represents a (non-negative) radius, the modulus can be leaved apart.2012-08-02