Problem — Given V find q

Problem

Given $V$, find $q$. An electron at a distance of $1\ \text{nm}$ from an unknown charge $q$ feels a potential $V = 1{,}44\ \text{V}$. Find $q$ and identify the particle.

Solution

Symbolic setup. The potential of a point charge is $V = k\,\dfrac{q}{r}$. Isolating $q$:

$q = \frac{V\,r}{k}$

Numerical substitution. With $V = 1{,}44\ \text{V}$, $r = 1\cdot 10^{-9}\ \text{m}$, $k = 8{,}99\cdot 10^9\ \text{N·m}^2/\text{C}^2$:

$q = \frac{1{,}44 \cdot 1\cdot 10^{-9}}{8{,}99\cdot 10^9} = \frac{1{,}44\cdot 10^{-9}}{8{,}99\cdot 10^9} \approx 1{,}60\cdot 10^{-19}\ \text{C}$

The value matches the elementary charge $e = 1{,}6\cdot 10^{-19}\ \text{C}$ and is positive (the potential detected is positive): the particle is a proton.

$\ev{q \approx 1{,}6\cdot 10^{-19}\ \text{C} = e\ (\text{proton})}$

Links

Topics: Electric field and potential Concepts: Electric potential · Electric charge