# Why the charge is zero in this problem? [closed]

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?

• It shouldn't be zero, right? Why do you think so? Oct 1 at 0:23
• @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 Oct 1 at 9:17

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.