Question

A point charge +Q is placed at point O as shown in the figure. Is the potential difference ${V_A} - {V_B}$ positive, negative or zero?

Answer 1

Hint: In this question use the concept that potential at any point is the multiplication of electric field at that point due to point charge +Q and the distance between the point charge and that point so use this concept to reach the solution of the question.

Complete Step-by-Step solution:
Let the distance between the point A and the point charge +Q be ${r_A}$ and the distance between the point B and the point charge +Q be ${r_B}$.
So as we see that
${r_B} > {r_A}$................... (1)
Now as we know that the electric field at point A due to point charge +Q is given as
$ \Rightarrow {E_A} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{r_A^2}}$
And the electric field at point A due to point charge +Q is given as
$ \Rightarrow {E_B} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{r_B^2}}$
Now as we know that the potential at any point is the multiplication of electric field at that point due to point charge +Q and the distance between the point charge and that point.
So the potential at point A is
$ \Rightarrow {V_A} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{r_A^2}} \times {r_A} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{{r_A}}}$
And the potential at point B is
\[ \Rightarrow {V_B} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{r_B^2}} \times {r_B} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{{r_B}}}\]
So the potential difference is
\[ \Rightarrow {V_A} - {V_B} = \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{{r_A}}} - \dfrac{1}{{4\pi {\varepsilon _o}}}\dfrac{Q}{{{r_B}}}\]
Now simplify the above equation we have,
\[ \Rightarrow {V_A} - {V_B} = \dfrac{Q}{{4\pi {\varepsilon _o}}}\left( {\dfrac{1}{{{r_A}}} - \dfrac{1}{{{r_B}}}} \right) = \dfrac{Q}{{4\pi {\varepsilon _o}}}\left( {\dfrac{{{r_B} - {r_A}}}{{{r_B}{r_A}}}} \right)\]
Now from equation (1) we know that ${r_B} > {r_A}$
Therefore, ${r_B} - {r_A} > 0$
\[ \Rightarrow \dfrac{Q}{{4\pi {\varepsilon _o}}}\left( {\dfrac{{{r_B} - {r_A}}}{{{r_B}{r_A}}}} \right) > 0\]
Therefore, \[{V_A} - {V_B} > 0\]
Hence \[{V_A} - {V_B}\] is positive.
So this is the required answer.

Note – Whenever we face such types of questions the always recall the formula of electric field and potential at any point and the relation between them which is all stated above so just substitute the these values in the given equation and simplify then check which distance is greater we will the required answer that is positive, negative or zero.