A Dipole Having A Moment $p = 3a_x - 5a_y + 10a_z \, \text{nC} \cdot \text{m}$ Is Located At $Q(1, 2, -4$\] In Free Space. Find $V$ At $P(2, 3, 4$\].

Introduction

In the realm of electromagnetism, the electric dipole is a fundamental concept that plays a crucial role in understanding various phenomena. A dipole is characterized by its dipole moment, which is a measure of the separation between two equal and opposite charges. In this article, we will delve into the calculation of the electric potential at a point in space due to a dipole located at a specific position. We will use the dipole moment and the position of the dipole to derive the electric potential at a given point.

The Dipole Moment

The dipole moment of a dipole is defined as the product of the magnitude of the charges and the distance between them. Mathematically, it can be represented as:

$$p = q \cdot d$$

where $p$ is the dipole moment, $q$ is the magnitude of the charges, and $d$ is the distance between them.

In this case, the dipole moment is given as $p = 3a_x - 5a_y + 10a_z \, \text{nC} \cdot \text{m}$. This means that the dipole has a magnitude of $13 \, \text{nC} \cdot \text{m}$ and is oriented in the $x$, $y$, and $z$ directions.

The Electric Potential Due to a Dipole

The electric potential due to a dipole at a point in space can be calculated using the following formula:

$$V = \frac{p \cdot r}{4 \pi \epsilon_0 r^3}$$

where $V$ is the electric potential, $p$ is the dipole moment, $r$ is the position vector of the point, and $\epsilon_0$ is the electric constant.

Calculating the Electric Potential

To calculate the electric potential at point $P(2, 3, 4)$ due to the dipole located at $Q(1, 2, -4)$, we need to first find the position vector of point $P$ relative to the dipole.

The position vector of point $P$ relative to the dipole is given by:

$$\vec{r} = (2 - 1)a_x + (3 - 2)a_y + (4 - (-4))a_z$$

$$\vec{r} = a_x + a_y + 8a_z$$

Now, we can calculate the electric potential using the formula:

$$V = \frac{p \cdot r}{4 \pi \epsilon_0 r^3}$$

Substituting the values, we get:

$$V = \frac{(3a_x - 5a_y + 10a_z) \cdot (a_x + a_y + 8a_z)}{4 \pi \epsilon_0 (1^2 + 1^2 + 8^2)}$$

$$V = \frac{(3 - 5 + 80)a_x + (0 - 5 + 0)a_y + (0 + 0 + 80)a_z}{4 \pi \epsilon_0 (1 + 1 + 64)}$$

$$V = \frac{78a_x - 5a_y + 80a_z}{4 \pi \epsilon_0 (66)}$$

$$V = \frac{78}{4 \pi \epsilon_0 (66)}a_x - \frac{5}{4 \pi \epsilon_0 (66)}a_y + \frac{80}{4 \pi \epsilon_0 (66)}a_z$$

Simplifying the Expression

To simplify the expression, we can use the fact that $\epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}$.

Substituting this value, we get:

$$V = \frac{78}{4 \pi (8.854 \times 10^{-12}) (66)}a_x - \frac{5}{4 \pi (8.854 \times 10^{-12}) (66)}a_y + \frac{80}{4 \pi (8.854 \times 10^{-12}) (66)}a_z$$

$$V = \frac{78}{1.144 \times 10^{-9}}a_x - \frac{5}{1.144 \times 10^{-9}}a_y + \frac{80}{1.144 \times 10^{-9}}a_z$$

$$V = 6.84 \times 10^9a_x - 4.37 \times 10^9a_y + 6.96 \times 10^9a_z$$

Conclusion

In this article, we calculated the electric potential at a point in space due to a dipole located at a specific position. We used the dipole moment and the position of the dipole to derive the electric potential at a given point. The electric potential was calculated using the formula $V = \frac{p \cdot r}{4 \pi \epsilon_0 r^3}$, and the result was simplified to obtain the final expression.

The electric potential due to a dipole is an important concept in electromagnetism, and it has numerous applications in various fields, including physics, engineering, and materials science. Understanding the electric potential due to a dipole is crucial for designing and analyzing various devices, such as antennas, transmitters, and receivers.

References

• [1] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
• [2] Griffiths, D. J. (2013). Introduction to Electrodynamics. Pearson Education.
• [3] Feynman, R. P. (1963). The Feynman Lectures on Physics. Addison-Wesley.

Note: The values used in the calculation are in SI units.

Introduction

In our previous article, we explored the concept of a dipole in free space and calculated the electric potential at a point in space due to a dipole located at a specific position. In this article, we will address some of the frequently asked questions related to the topic.

Q: What is a dipole?

A: A dipole is a pair of equal and opposite charges separated by a small distance. It is a fundamental concept in electromagnetism and is used to describe various phenomena, including electric fields, magnetic fields, and electromagnetic waves.

Q: What is the dipole moment?

A: The dipole moment is a measure of the separation between two equal and opposite charges. It is defined as the product of the magnitude of the charges and the distance between them. Mathematically, it can be represented as:

$$p = q \cdot d$$

Q: How is the electric potential due to a dipole calculated?

A: The electric potential due to a dipole at a point in space can be calculated using the following formula:

$$V = \frac{p \cdot r}{4 \pi \epsilon_0 r^3}$$

where $V$ is the electric potential, $p$ is the dipole moment, $r$ is the position vector of the point, and $\epsilon_0$ is the electric constant.

Q: What is the significance of the dipole moment in electromagnetism?

A: The dipole moment is a crucial concept in electromagnetism, as it describes the orientation and strength of a dipole. It is used to calculate the electric field, magnetic field, and electromagnetic waves. The dipole moment is also used to describe the behavior of atoms and molecules in various physical and chemical processes.

Q: Can a dipole be used to describe the behavior of a single charge?

A: No, a dipole cannot be used to describe the behavior of a single charge. A dipole is a pair of equal and opposite charges, whereas a single charge is a point charge. The electric potential due to a single charge is given by:

$$V = \frac{q}{4 \pi \epsilon_0 r}$$

Q: How does the electric potential due to a dipole change with distance?

A: The electric potential due to a dipole decreases with distance. As the distance between the dipole and the point increases, the electric potential decreases. This is because the dipole moment is inversely proportional to the cube of the distance.

Q: Can the electric potential due to a dipole be negative?

A: Yes, the electric potential due to a dipole can be negative. This occurs when the dipole moment and the position vector of the point are in opposite directions.

Q: What is the relationship between the electric potential due to a dipole and the electric field?

A: The electric potential due to a dipole is related to the electric field by the following equation:

$$E = -\nabla V$$

where $E$ is the electric field and $V$ is the electric potential.

Q: Can the electric potential due to a dipole be used to describe the behavior of a conductor?

A: Yes, the electric potential due to a dipole can be used to describe the behavior of a conductor. The electric potential due to a dipole can induce charges on the surface of a conductor, which can lead to various physical and chemical processes.

Conclusion

In this article, we addressed some of the frequently asked questions related to the topic of a dipole in free space. We explored the concept of a dipole, the dipole moment, and the electric potential due to a dipole. We also discussed the significance of the dipole moment in electromagnetism and its relationship with the electric field.

References

• [1] Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
• [2] Griffiths, D. J. (2013). Introduction to Electrodynamics. Pearson Education.
• [3] Feynman, R. P. (1963). The Feynman Lectures on Physics. Addison-Wesley.

Note: The values used in the calculation are in SI units.