Question

What is the total charge on an infinite wire carring a charge density

λ =

1

|x|+α where α is a cosntant.

Answer 1

Given,

Charge density of the charged wire $(\lambda) = \frac{1}{|x|+\alpha}$

$\Rightarrow dq=\lambda dx$

$\Rightarrow \int dq = \int \lambda dx$

Now, let the total length of the wire be l. So, integration limit will be $-l$ to $+l$

$\Rightarrow \int_{q=0}^{q=Q} dq=\int_{-l}^{l}\lambda dx$

Now, taking the integration

$Q=\int_{x=0}^{x=l} \frac{dx}{x-\alpha}-\int_{x=0}^{x=l} \frac{dx}{x+\alpha}$

$\Rightarrow Q = [\ln(x-\alpha)]_{0}^{l}-\ln{(x+\alpha)}]_{0}^{l}$

$\Rightarrow Q = \ln(x-\alpha)-ln(x+\alpha)$

$\Rightarrow Q = \ln{\frac{(x-\alpha)}{(x+\alpha)}}$

$\Rightarrow Q = \ln(\frac {x(1-\frac{\alpha}{x})}{x(1+\frac{\alpha}{x})})$

$x\rightarrow \infty$

$\Rightarrow Q = \ln 1\sim 0$

Hence, net charge on the straight infinite wire be zero.