Two parallel infinite opposite charged lines (of uniform density $\lambda$) are placed at distance $L$ from each other. Need to calculate the electric dipole moment of this system.
How do you do such a calculation? do I need to sum the moment dipoles of infinite tiny charges?
The dipole moment of an arbitrary charge density distribution $\rho(\mathbf r)$ is defined as $$ \mathbf d = \int \mathbf r \, \rho(\mathbf r) \:\!\mathrm d\mathbf r, $$ where $\rho(\mathbf r) \:\!\mathrm d\mathbf r$ needs to be understood as a singular measure (say, using suitable delta-function distributions) in the case of point, surface or line charges. For the example of a line charge parametrized by $\mathbf r(s)$ over some parameter $s\in (a,b)$ and with longitudinal charge density $\lambda(s)$, this reads $$ \mathbf d = \int_a^b \mathbf r(s) \:\! \lambda(s) \:\!\mathrm ds. $$ If you have two infinite straight line charges at separation $a$ with constant longitudinal charge density $\lambda$ and with orthogonal unit separation vector $\mathbf u$, then this calculation yields a constant dipole moment $\Delta \mathbf p = \lambda \:\!\Delta L \:\! a\mathbf u$ per stretch of length $\Delta L$ of the two wires. The total dipole moment of the system can be inferred from that.
