Derivative of the retarded vector potential

In summary, the given conversation discusses the potential vector for an oscillating electric dipole, including its expression and components such as direction, current, distance, and dipole moment. It also mentions using the Lorentz Gauge and the chain rule to find the derivative of the potential. The final result is expressed as a function of time, distance, and the dipole's components.
  • #1
Salmone
101
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In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis## where the dipole is oscillating, ##I_0## is the current (##I(t)=I_0cos(\omega t)##), ##d## is the distance between the charges of the dipole and ##r## is the distance between the origin of the system and the point where I want to calculate the potential vector. Let ##\vec{p}=\hat{k}qd=\frac{\hat{k}dI_0}{\omega}sin(\omega t)## be the dipole moment, it is possible to rewrite the potential vector as ##\vec{A}=\frac{\mu_0}{4\pi}\frac{\vec{\dot p(t-r/c)}}{r}## where ##\vec{\dot p}## is the derivative with respect to time.
If we use Lorentz Gauge we have ##-\epsilon_0\mu_0\frac{\partial V}{\partial t}=\vec{\nabla} \cdot \vec{A}##, since ##\vec{A}## is directed along the ##z-axis## we have ##div \vec{A}=\frac{\partial A}{\partial z}## so ##\frac{\partial V}{\partial t}=-\frac{1}{\epsilon_0 \mu_0}\frac{\partial A}{\partial z}##.

The question: how do you do this last derivative?

The result is: ##\frac{\partial V}{\partial t}=\frac{1}{4\pi\epsilon_0} \left( \frac{- \ddot p(t-r/c)}{cr}+\frac{\dot p(t-r/c)}{r^2} \right)\frac{z}{r}##
 
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  • #2
When you took calculus, did you learn the chain rule?
 
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Related to Derivative of the retarded vector potential

1. What is the definition of the derivative of the retarded vector potential?

The derivative of the retarded vector potential is a mathematical concept used in electromagnetism to describe the change in the electromagnetic field over time. It is calculated by taking the time derivative of the retarded vector potential, which is a function of the electric and magnetic fields at a given point in space and time.

2. How is the derivative of the retarded vector potential used in electromagnetism?

The derivative of the retarded vector potential is used to calculate the electric and magnetic fields at a given point in space and time. It is an important tool in understanding the behavior of electromagnetic waves and their interactions with charged particles.

3. Can the derivative of the retarded vector potential be negative?

Yes, the derivative of the retarded vector potential can be negative. This indicates a decrease in the electric or magnetic field over time. It is important to note that the magnitude and direction of the derivative can vary depending on the specific situation and should be interpreted in context.

4. How is the derivative of the retarded vector potential related to the speed of light?

The derivative of the retarded vector potential is directly related to the speed of light. This is because the retarded vector potential is based on the concept of causality, meaning that the electric and magnetic fields at a given point in space and time are influenced by the fields at previous points in time. The speed of light represents the maximum speed at which these influences can travel.

5. Are there any practical applications of the derivative of the retarded vector potential?

Yes, the derivative of the retarded vector potential has several practical applications. It is commonly used in the design and analysis of electromagnetic devices, such as antennas and circuitry. It is also used in the study of radiation from accelerating charges, which has important implications in fields such as radio astronomy and medical imaging.

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