# Electric dipole moment of two infinite charged lines

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?

• Do you mean the electric dipole moment? Are the two lines oppositely charged? Apr 12 '19 at 16:39
• yes, sorry I didn't mention Apr 12 '19 at 16:43

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.