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How to calculate force between two charges

In this article, we will learn how to calculate the force between two charges. The force between two charges can be found using Coulomb’s Law in electrostatics. Before going any further, let us learn what is Coulomb’s Law.

Coulomb’s Law states that

The electrostatic force of interaction between two point charges is directly proportional to the product of the charges, inversely proportional to the square of the distance between them and acts along the line joining the two charges.

Coulomb’s Law Statement

$\rightarrow$ Coulomb’s Law applies to stationary point charges.

To represent it mathematically, consider two point charges $q_a$ and $q_b$ separated by a distance $r$.

Let $F$ be the electrostatic force between these two charges. According to Coulomb’s Law

  1. $F\propto q_a q_b$
  2. $F\propto \frac{1}{r^{2}}$

Combining these we get

$F=k\frac{q_a q_b}{r^2}$

where $k$ is the constant of proportionality called electrostatic force constant. In SI units
$$k=\frac{1}{4\pi\epsilon_0}\approx9\times 10^9Nm^2C^{-2}$$
where $\epsilon_0$ is the permittivity of free space of vacuum.

Let us now look at how to calculate the force between two charges. For this, consider this question:-

Solved Question

Two positive charges of $3\times10^{-4}C$ and $8.0\times10^{-4}C$ are separated by $0.30m$. What is the force between these two charges?

Step 1 – Analyze and sketch the problem

  • With the help of a diagram to show charges and the distance between them.
  • Label the force acting between the charges

Step 2 – Write down the known and unknown quantities as given in the question

Known Quantities
$q_a=3.0\times10^{-4}C$
$q_b=8.0\times10^{-4}C$
$r=0.30m$
Unknown Quantity
$F_{\text{b on a}}=?$

Step 3 – Solve the unknown
We have to find the force between the two charges in this problem. So, force due to charge $b$ to charge $a$ would be
$F_{\text{b on a}}=k\frac{q_aq_b}{r^2}$
Plugging in the values of $q_a$, $q_b$ and $r$
$F_{\text{b on a}}=(9\times 10^9Nm^2C^{-2})\frac{(3.0\times10^{-4}C)(8.0\times10^{-4}C)}{(0.30m)^2}$
On calculating it we get
$F_{\text{b on a}}=2.4\times10^4N$


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