Potential vector of a oscilating dipole:

In summary, the conversation discusses the reasoning behind the derivation of the electric potential of an oscillating dipole, as presented in Griffiths' Electrodynamics book. The conversation also explores the use of the chain rule and the Cartesian coordinates method to calculate the components of ∇[r̂⋅⃗ṗ(t0)/r]. The result is a simplified expression that involves the derivative of the acceleration of the dipole with respect to the change in time.
  • #1
LCSphysicist
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Homework Statement
I.
Relevant Equations
.
I am passing through some difficulties to understand the reasoning to derive the electric potential of an oscilating dipole used by Griffths at his Electrodynamics book:
Knowing that ##t_o = t - r/c##,
Captura de tela de 2022-05-13 07-55-47.png

What exactly he has used here to go from the first term after "and hence" to the second term? (The one involving ##\ddot p##). Maybe ##\nabla = \nabla t_o \frac{d}{dt_o}## ? But this does not make sense.
 
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  • #2
A straightforward, but somewhat tedious, method is to work in cartesian coordinates.

Consider the ##x##-component of ##\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]##:

$$\left(\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]\right)_x = \frac{\partial}{\partial x} \left[ \frac{x}{r^2} \dot p_x(t_0) + \frac{y}{r^2} \dot p_y(t_0) +\frac{z}{r^2} \dot p_z(t_0) \right]$$
Expand this out. You can drop terms that fall off faster than ##1/r##.

When calculating ##\large \frac{\partial \dot p_x(t_0)}{\partial x}##, recall that ##t_0 = t-r/c =t-\sqrt{x^2+y^2+z^2}/c##. So, you can use the chain rule to write $$ \frac{\partial \dot p_x(t_0)}{\partial x} = \frac{d \dot p_x}{dt_0} \frac{\partial t_0}{\partial x}$$
Likewise for dealing with ##\large \frac{\partial \dot p_y(t_0)}{\partial x}## and ##\large \frac{\partial \dot p_z(t_0)}{\partial x}##.

You should find that
$$\left(\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]\right)_x = \left( \frac {\mathbf{\hat{r}}}{r} \cdot \mathbf{\ddot p}(t_0)\right) \frac{\partial t_0}{\partial x}$$
Similar results are obtained for the ##y## and ##z## components of ##\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]##.
 
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  • #3
Herculi said:
Homework Statement:: I.
Relevant Equations:: .

Maybe ∇=∇toddto ? But this does not make sense.
Yes it does. That is just the chain rule:
$$
[\nabla \dot{\vec p}(t_0)]_{ij} = \frac{\partial \dot p^j(t_0)}{\partial x^i} = \frac{d \dot p^j}{dt_0}\frac{\partial t_0}{\partial x^i} = \ddot{p}^j(t_0) \partial_i t_0.
$$
 
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Related to Potential vector of a oscilating dipole:

What is the potential vector of an oscillating dipole?

The potential vector of an oscillating dipole refers to the mathematical representation of the electric potential and magnetic potential fields created by a dipole that is undergoing oscillations.

How is the potential vector of an oscillating dipole calculated?

The potential vector of an oscillating dipole can be calculated using the equations for the electric potential and magnetic potential fields, which take into account the amplitude, frequency, and orientation of the dipole's oscillations.

What factors affect the potential vector of an oscillating dipole?

The potential vector of an oscillating dipole is affected by the amplitude, frequency, and orientation of the dipole's oscillations, as well as the distance from the dipole and the surrounding medium's electrical and magnetic properties.

What is the significance of the potential vector of an oscillating dipole?

The potential vector of an oscillating dipole is significant because it helps us understand and predict the behavior of electromagnetic waves, which are created by oscillating dipoles. It also has practical applications in various fields, such as telecommunications and medical imaging.

How does the potential vector of an oscillating dipole relate to the concept of radiation?

The potential vector of an oscillating dipole is closely related to the concept of radiation, as it describes the electromagnetic fields that are radiated by the dipole as it undergoes oscillations. This radiation is responsible for the propagation of electromagnetic waves through space.

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