Two point charges, \(1 \mathrm{nC}\) at \((0,0,0.1)\) and \(-1 \mathrm{nC}\) at \((0,0,-0.1)\), are in free space. \((a)\) Calculate \(V\) at \(P(0.3,0,0.4) .\) (b) Calculate \(|\mathbf{E}|\) at \(P .(c)\) Now treat the two charges as a dipole at the origin and find \(V\) at \(P\).

Understanding the electric potential due to point charges is a fundamental aspect of electromagnetism. When we talk about the electric potential brought on by a point charge, we're referring to the work done per unit charge to move a small positive test charge from infinity to a point in the vicinity of a source charge, without causing any acceleration.

Mathematically, the electric potential (\( V \) due to a point charge is given by the equation: \[\begin{equation} V = \frac{kQ}{r} \end{equation}\] where:

• \textbf{k} is the Coulomb's constant (\textbf{8.99 \times 10^9 N m^2/C^2}),
• \textbf{Q} is the charge,
• and \textbf{r} is the distance from the point charge to the point where the potential is being measured.

If we have multiple charges, as in our example with two point charges, the total electric potential at a certain point in space is the algebraic sum of the potentials due to each charge separately. It is important to note that because electric potential is a scalar quantity, it does not have a direction and can simply be summed or subtracted depending on the sign of each charge's potential contribution.

When solving for the electric potential at a point due to multiple charges, carefully calculate the distance from each charge to the point of interest, and apply the formula for each charge's contribution. The sum of these contributions gives the total electric potential at that point.

The electric field magnitude represents a vector field around charged objects, indicating the force that a positive test charge would experience at any point in space. To calculate the electric field magnitude, we use the formula: \[\begin{equation}|\textbf{E}| = \frac{kQ}{r^2}\end{equation}\]where

• \textbf{k} is Coulomb's constant,
• \textbf{Q} is the charge causing the field,
• and \textbf{r} is the distance from the charge to the point where we're calculating the field.

To determine the total electric field caused by multiple charges, calculate the vector sum of the fields due to each charge. This process often involves vector decomposition, vector addition, and the use of trigonometric identities.

Keep in mind that electric field lines point away from positive charges and toward negative charges. The magnitude of the electric field gives a sense of how strong the electric field is at that point but not its direction. To fully understand the electric field at a point, you need to consider both the magnitude and the direction to know the force that would act on a positive test charge placed there.

An electric dipole consists of two point charges of equal magnitude but opposite sign, separated by a small distance. The concept of electric dipole potential is vital to understanding the resulting electric field and potential patterns due to this configuration.

The potential (\textbf{V}) at a point in space due to a dipole is calculated using the formula: \[\begin{equation}V_{dipole} = \frac{k \boldsymbol{p} \cdot \boldsymbol{r}}{r^3}\end{equation}\]here

• \textbf{k} is Coulomb's constant,
• \textbf{\boldsymbol{p}} is the electric dipole moment, which is the product of the charge and the vector distance between the charges (\textbf{Qd}),
• and \textbf{\boldsymbol{r}} is the position vector of the point from the dipole's center.

Analyzing the potential of a dipole requires understanding vectors and their dot product, as the formula involves the cosine of the angle between the dipole moment and the position vector. This approach simplifies the calculation of electric potential when dealing with systems that can be approximated as dipoles, which is often the case in molecular and cellular biology, as well as in various fields of nanotechnology and materials science.