Question

A test charge $q$ is located $1m$ from a much larger and stationary charge $Q$. While at this location, the test charge, $q$ experiences a force of $F$ from the stationary charge $Q$. The test charge is then moved to a new location $2m$ from $Q$ . What force will the test charge $q$ experience from the stationary charge $Q$ at the new location?
A.)\[\left( {\dfrac{1}{4}} \right)F\]
B.)$\left( {\dfrac{1}{2}} \right)F$
C.)$F$
D.)\[2F\]

Answer 1

Hint- You can start by describing electromagnetic force. Then you can describe Coulomb’s law and its equation. Use Coulomb’s law for both the initial and final conditions given in the problem, compare the two forces and find the solution.

Complete step-by-step answer:
Given in the problem,
$
  {r_i} = 1m \\
  {r_f} = 2m \\
$

Let the initial and final force be ${F_i}$and${F_f}$

According to Coulomb’s law

$F = \dfrac{{k{q_1}{q_2}}}{{{r^2}}}$
Here,

$k = $Proportionality constant
$F = $Electrostatic force
${q_1} = $1st charge
${q_2} = $2nd charge
$r = $Distance between the two charges

Applying Coulomb’s law for initial conditions

${F_i} = \dfrac{{kQq}}{{r_i^2}}$
${F_1} = \dfrac{{kQq}}{{{{(1)}^2}}}$(Equation 1)

Applying Coulomb’s law for final conditions

${F_f} = \dfrac{{kQq}}{{r_f^2}}$
$ \Rightarrow {F_f} = \dfrac{{kQq}}{{{{\left( 2 \right)}^2}}}$
$ \Rightarrow {F_f} = \dfrac{{kQq}}{4}$
$ \Rightarrow {F_f} = \dfrac{{{F_i}}}{4}$

Hence, option A is the correct choice

Additional Information:
The force that a charge experiences in the presence of another charge is known as electrostatic force. This force is also known as Coulomb force, after the name of its discoverer Charles-Augustin de Coulomb.

The charge experiences Coulomb’s force due to the formation of an electric field around the other charge. Imagine a fisherman entrapping a fish in its net and pulling it, this is somewhat how electric fields trap a charge.

Note: In the solution, we considered both $Q$ and to be positive, but, you can also consider charges negative. Changing the nature of charges will not affect the final answer, and it only means that the force can be attractive in case of opposite nature or it can be repulsive.