Chapter 3.1, Problem 2PPC

Interpretation Introduction

Interpretation:

The electrostatic potential energies between the given charges in the given statement should be calculated by using the concept of electrostatic energy.

Concept Introduction:

Energy is the ability to do work or transfer heat where work is the movement of a body using some force.  The SI unit of energy is joule ($\text{J}$).  Energy can be in the form of kinetic energy or potential energy.  Potential energy is the energy possessed by virtue of position. Other forms of potential energy are chemical energy and electrostatic energy.

Electrostatic energy is potential energy which results from the interaction of charged particles.  Oppositely charged particles attract each other and particles of like charges repel each other.  The magnitude of the resulting electrostatic potential energy is proportional to the product of the two charges (${\text{Q}}_{\text{1}}{\text{ and Q}}_{\text{2}}$) divided by the distance between the two charges ($\text{d}$).

${\text{E}}_{\text{el}}\propto \frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{d}}$

If the charges ${\text{Q}}_{\text{1}}{\text{ and Q}}_{\text{2}}$ are opposite (i.e., one positive and one negative), ${\text{E}}_{\text{el}}$ gets a negative value which indicates attraction.  Like charges (i.e., either both positive or both negative) result in a positive value for ${\text{E}}_{\text{el}}$ which indicates repulsion.

To calculate: Determine the ${\text{E}}_{\text{el}}$ values between the given charges

Explanation of Solution

(i)

The electrostatic energy between two charges is calculated using the formulae: ${\text{E}}_{\text{el}}\propto \frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{d}}$

For charges of +2 and −2, separated by a distance of $\text{d}$, the electrostatic potential energy is ${\text{E}}_{\text{el}}$.  Both the distance and the proportionality constant cancel in the formula.

For charges +2 and −2:

$\begin{array}{l}{\text{E}}_{\text{el(+2,}-\text{2)}}=\text{ c}\frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{2d}}{\text{, Q}}_{\text{1}}={\text{ +2 and Q}}_{\text{2}}=-\text{2}\\ \text{E }=\text{ c}\frac{\text{2 × }\left(-\text{2}\right)}{\text{2d}}\\ \text{Skip the negative sign, c and d,}\\ \text{E }=\frac{\text{2}}{\text{1}}\\ \frac{\text{1}}{\text{2}}\text{E}\end{array}$

Explanation:

(ii)

For charges +1 and −4:

$\begin{array}{l}{\text{E}}_{\text{el(+1,}-\text{4)}}=\text{ c}\frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{2d}}{\text{, Q}}_{\text{1}}={\text{ +1 and Q}}_{\text{2}}=-\text{4}\\ \text{E }=\text{ c}\frac{\text{1 × }\left(-\text{4}\right)}{\text{2d}}\\ \text{Skip the negative sign, c and d,}\\ \text{E }=\frac{\text{2}}{\text{1}}\\ \frac{\text{1}}{\text{2}}\text{E}\end{array}$

Explanation:

(iii)

For charges +3 and −4:

$\begin{array}{l}{\text{E}}_{\text{el(+3,}-\text{4)}}=\text{ c}\frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{d}}{\text{, Q}}_{\text{1}}={\text{ +3 and Q}}_{\text{2}}=-\text{4}\\ \text{E }=\text{ c}\frac{\text{3 × }\left(-\text{4}\right)}{\text{d}}\\ \text{Skip the negative sign, c and d,}\\ \text{E }=\frac{\text{12}}{\text{1}}\\ \frac{\text{1}}{\text{12}}\text{E}\end{array}$

Explanation:

(iv)

For charges +4 and −2:

$\begin{array}{l}{\text{E}}_{\text{el(+4,}-\text{2)}}=\text{ c}\frac{{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}}{\text{3d}}{\text{, Q}}_{\text{1}}={\text{ +4 and Q}}_{\text{2}}=-\text{2}\\ \text{E }=\text{ c}\frac{\text{4 × }\left(-\text{2}\right)}{\text{3d}}\\ \text{Skip the negative sign, c and d,}\\ \text{E }=\frac{\text{6}}{\text{3}}\\ \frac{\text{3}}{\text{6}}\text{E}\\ \frac{\text{1}}{\text{2}}\text{E}\end{array}$

Therefore, the electrostatic potential energy between charges of $\text{+2 and }-\text{2 is }\frac{\text{1}}{2}\text{E}$.  The electrostatic potential energy between charges of $\text{+1 and }-\text{4 is }\frac{\text{1}}{2}\text{E}$ .  The electrostatic potential energy between charges of $\text{+3 and }-\text{4 is }\frac{\text{1}}{12}\text{E}$.  The electrostatic potential energy between charges of $\text{+4 and }-\text{2 is }\frac{\text{1}}{2}\text{E}$.

Conclusion

The electrostatic potential energies between the given charges in the given statement are calculated by using the concept of electrostatic energy.