Say we have a bar centered on the origin, orientated with the z-axis, such that $a$ is the measure of the bar and it has a linear density that varies within $z$ with the following expression $\lambda(z)=\lambda_0 z$. We'd have then $\frac{a}{2}$ of the bar in the positive z axis and $\frac{a}{2}$ of the bar in the negative z axis, so if we wanted to find the charge we'd just do $Q=\int_{-a/2}^{a/2} \lambda(z) dz$ so that we get: $$Q=\lambda_0 \int_{-a/2}{a/2} zdz=\lambda_0 \frac{z^2}{2}|_{-a/2}^{a/2}=0$$ why do we get zero? It shouldn't be zero, right? What is wrong?
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4$\begingroup$ It shouldn't be zero, right? Why do you think so? $\endgroup$– GhosterOct 1 at 0:23
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$\begingroup$ @Ghoster I don't know, I'm asked to find the multipolar expansion of this and I don't know how to do it if the charge is 0 $\endgroup$– UlshyOct 1 at 9:17
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1 Answer
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You shoud get zero because you assume $\lambda(z)=\lambda_{0} z$ and therefore $\lambda(z) < 0$ when $z <0$ and vice-versa. Since your integral is symmetic w.r.t. $z$ so the final result becomes zero.
