Question

Two point charges \(-q\) and \(+q\) are located at points \((0,0,-a)\) and \((0,0, a)\) respectively. The potential at a point \((0,0, z)\) where \(z>a\) is \(\ldots \ldots\) (A) \(\left[(2 \mathrm{q} a) /\left\\{4 \pi \epsilon_{0}\left(z^{2}+a^{2}\right)\right\\}\right]\) (B) \(\left[\mathrm{q} /\left(4 \pi \epsilon_{0} \mathrm{a}\right)\right]\) (C) \(\left[\right.\) (qa) \(\left./\left(4 \pi \in_{0} z^{2}\right)\right]\) (D) \(\left[(2 q a) /\left\\{4 \pi \epsilon_{0}\left(z^{2}-a^{2}\right)\right\\}\right]\)

Short answer

The correct answer for the electric potential at point \((0,0,z)\) where \(z>a\) should be 0, as the potentials due to the \(-q\) and \(q\) charges cancel each other out. However, none of the given options match this result, indicating a possible error in the options provided.

Step-by-step solution

Calculate the distance to each point charge

Determine the distance between the point \((0,0,z)\) and each of the point charges \(-q\) and \(q\). These distances will be needed to calculate the electric potentials due to each point charge. Distance to \(-q\): \(r_{1}=|z-(-a)|=z+a\) Distance to \(+q\): \(r_{2}=|z-a|\)

Calculate the electric potentials due to each point charge

Now, we will calculate the electric potential at point \((0,0,z)\) due to each point charge using the formula: \[V = \frac{Q}{4\pi\epsilon_{0}r}\] Electric potential due to \(-q\): \[V_{1} = \frac{-q}{4\pi\epsilon_{0}(z+a)}\] Electric potential due to \(q\): \[V_{2} = \frac{q}{4\pi\epsilon_{0}(z-a)}\]

Add the electric potentials

Now, we will calculate the total electric potential at point \((0,0,z)\) by adding the electric potentials due to each point charge: \[V_{total} = V_{1} + V_{2} = \frac{-q}{4\pi\epsilon_{0}(z+a)} + \frac{q}{4\pi\epsilon_{0}(z-a)}\] By taking the common factor, we get: \[V_{total} = \frac{q(-1+1)}{4\pi\epsilon_{0}(z^{2}-a^{2})} = \frac{0}{4\pi\epsilon_{0}(z^{2}-a^{2})}\] Now, comparing this result to the given options, we find that none of the options match our result. Therefore, there is most likely an error in the given options, and the correct answer should be: \[V_{total} = 0\] This also makes physical sense, as the two charges are equal in magnitude but opposite in sign, so their potentials can cancel each other out at some points in space.

Key concepts

Point Charges

Understanding point charges is crucial in electrostatics. A point charge is an electric charge located at a single point in space. It's an idealized model that helps simplify calculations.

Key features include:

• They allow the application of theoretical principles to practical problems.
• Assume the charge occupies no space and is concentrated at a point.

In the given exercise, we deal with two point charges \( -q \, \ +q \,\) located symmetrically around the origin. They make calculations involving electric fields and potentials easier because of their simplicity.

Coulomb's Law

Coulomb's Law lays the groundwork for understanding interactions between point charges. It defines the electric force between two stationary charges. The formula is expressed as:

\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]

where \( F \, \ k \, \ q_1 \, \ q_2 \,\) and \( r \,\) are the force, Coulomb's constant, the charges, and the distance between them, respectively.

Insights include:

• The forces are attractive if the charges are of opposite signs and repulsive if the same.
• The force is inversely proportional to the square of the distance, highlighting quicker dissipation over distance.

In the exercise, Coulomb's Law facilitates finding the electric potentials by considering the distances to each charge.

Electrostatics

Electrostatics focuses on the behavior of electric charges at rest. It is a fundamental part of understanding electric fields and potentials. In the scenario with two point charges, the principles of electrostatics help explain how these charges interact to produce a net electric potential.

Main ideas include:

• Charges exert forces on each other.
• The electric potential is the energy required to move a charge within an electric field.

In this particular exercise, electrostatic principles guide the addition of potentials from both charges, leading to an exploration of their cancellation.

Electric Fields

Electric fields describe how a charge exerts a force on other charges in its vicinity. They provide a way to visualize the influence of charges and are represented by vectors radiating from positive charges and converging on negative charges.

Key concepts of electric fields include:

• The electric field \( E \,\) at a point is given by \( E = \frac{F}{q} \,\), where \( F \,\) is the force and \( q \,\) the charge.
• The field's effect diminishes with increased distance from the charge.

In the context of the exercise, the electric field results from the combination of fields due to each point charge, contributing to the zero net electric potential at certain points.