Problem — Two charges on a rod

Problem

Two charges on a rod. Two charges $+3\ \mu\text{C}$ and $-2\ \mu\text{C}$ at the ends of an insulating rod of length $30\ \text{cm}$. Calculate the magnitude and direction of $\vv{E}$ at the midpoint of the rod.

Solution

Geometry. The midpoint $P$ is a distance $r = 0{,}15\ \text{m}$ from each charge. We apply the superposition principle: $\vv{E} = \vv{E}_1 + \vv{E}_2$.

Field of the $+3\ \mu\text{C}$ charge. Points away from the positive charge, i.e. towards the negative end:

$E_1 = k\,\frac{q_1}{r^2} = 8{,}99\cdot 10^9 \cdot \frac{3\cdot 10^{-6}}{(0{,}15)^2} = \frac{2{,}697\cdot 10^4}{0{,}0225} \approx 1{,}20\cdot 10^6\ \text{V/m}$

Field of the $-2\ \mu\text{C}$ charge. Points towards the negative charge, i.e. in the same direction as $\vv{E}_1$:

$E_2 = k\,\frac{q_2}{r^2} = 8{,}99\cdot 10^9 \cdot \frac{2\cdot 10^{-6}}{0{,}0225} = \frac{1{,}798\cdot 10^4}{0{,}0225} \approx 7{,}99\cdot 10^5\ \text{V/m}$

Superposition. The two contributions have the same direction (both point towards the $-2\ \mu\text{C}$ end), so they add:

$E = E_1 + E_2 = 1{,}20\cdot 10^6 + 0{,}80\cdot 10^6 \approx 2{,}0\cdot 10^6\ \text{V/m}$

$\ev{E \approx 2{,}0\cdot 10^6\ \text{V/m, verso la carica negativa}}$

Linked atoms

Topics: Campo elettrico e potenziale Concepts: Campo elettrico Skills: Principio di sovrapposizione