Question

Two parallel infinite line charges with linear charge densities + λ C/m and - λ C/m are placed at a distance of 2R in free space. What is the electric field mid-way between the two line charges?
(1) zero
(2) 2λ/(πε₀R) N/C
(3) λ/(πε₀R) N/C
(4) λ/(2πε₀R) N/C

Answer 1

The electric field mid-way between the two line charges is $\bold{E = \frac{\lambda }{\pi\varepsilon _{o} R}\widehat{i}N/C}$

Step by step explanation:

The electric field in between lines can be calculated by the following formula.

                       $\bold{E = \frac{\lambda }{2\pi\varepsilon _{o} R}\widehat{i}N/C}$

From the given,

Two parallel infinite line charges with linear charge densities + λ C/m and - λ C/m

The distance between two lines = 2R

Electric field due to line charge (1)

                           $\bold{E _{1}= \frac{\lambda }{2\pi\varepsilon _{o} R}\widehat{i}N/C}$

Electric field due to line charge (2)

                           $\bold{E _{2}= \frac{\lambda }{2\pi\varepsilon _{o} R}\widehat{i}N/C}$

The net electrical field can be calculated by the following formula.

                            $\bold{\overrightarrow{E}_{net}=\overrightarrow{E_{1}}+\overrightarrow{E_{2}}}$..........................(1)

Substitute the $E_{1}\,and\,E_{2}$ values in the equation (1)

                                   $\Rightarrow \frac{\lambda }{2\pi\varepsilon _{o} R}\widehat{i}N/C+\frac{\lambda }{2\pi\varepsilon _{o} R}\widehat{i}N/C$

                                   $\Rightarrow \frac{\lambda }{\pi\varepsilon _{o} R}\widehat{i}N/C$

Therefore, The electric field mid-way between the two line charges is $\bold{E = \frac{\lambda }{\pi\varepsilon _{o} R}\widehat{i}N/C}$

Hence, correct answer - 3