Something about retarded potentials for oscillating electric dipole

In summary, the potential vector at a point where an electric dipole is oscillating can be found using the following equation:
  • #1
Salmone
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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.
 
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  • #2
You can start with the source of a point charge,
$$\rho(t,\vec{x})=q \delta^{(3)}[\vec{x}-\vec{y}(t)], \quad \vec{j}(t,\vec{x})= q \dot{\vec{y}}(t) \delta^{(3)}[\vec{x}-\vec{y}(t)],$$
where ##\vec{y}(t)## is the trajectory of the charge.

Now we assume that
$$\vec{y}(t)=\vec{d} \sin(\omega t).$$
For ##r=|\vec{x}|\gg |\vec{d}|## we can expand the charge-current distribution up to first order in ##\vec{d}##,
$$\rho(t,\vec{x})=q \delta^{(3)}(\vec{x}) - q \vec{y}(t) \cdot \vec{\nabla} \delta^{(3)}(\vec{x}) + \mathcal{O}(\vec{d}^2), \quad \vec{j}(t,\vec{x})=q \dot{\vec{y}}(t) \delta^{(3)}(\vec{x}) + \mathcal{O}(\vec{d}^2).$$
From the first term of ##\rho## (of order ##\mathcal{O}(d^0)##) you get the electrostatic Coulomb field of a charge at rest in the origin (which you can easily verify using the retarded potential too).

For the terms of order ##\mathcal{O}(d)## you get for ##\vec{A}## (in SI units)
$$\vec{A}(t,\vec{x})=\frac{\mu_0}{4 \pi} \int_{\mathbb{R}^3} \mathrm{d}^3 x' \vec{j}(t-|\vec{x}-\vec{x}'|/c,\vec{x}') \frac{1}{|\vec{x}-\vec{x}'|}=\frac{\mu_0 q \omega \vec{d}}{4 \pi} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \cos[\omega (t-|\vec{x}-\vec{x}'|/c] \frac{\delta^{(3)}(\vec{x}')}{|\vec{x}-\vec{x}'|} = \frac{\mu_0 q \omega \vec{d}}{4 \pi r} \cos[\omega (t-r/c)].$$
With ##I_0=q \omega## that's the solution you are looking for.

The final equation, of course, must read
$$\vec{A}(t),\vec{x})=\frac{\mu_0}{4 \pi} \frac{\dot{\vec{P}}(t-r/c)}{r}.$$
 
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Related to Something about retarded potentials for oscillating electric dipole

1. What are retarded potentials for oscillating electric dipole?

Retarded potentials refer to the electromagnetic fields produced by a moving charged particle or an oscillating electric dipole. These potentials are calculated using Maxwell's equations and take into account the time delay in the propagation of electromagnetic waves.

2. How do retarded potentials affect an oscillating electric dipole?

Retarded potentials play a crucial role in the behavior of an oscillating electric dipole. They determine the strength and direction of the electric and magnetic fields around the dipole, which in turn affect its radiation pattern and energy dissipation.

3. What is the significance of retarded potentials in electromagnetism?

Retarded potentials are important in understanding the fundamental principles of electromagnetism and the behavior of electromagnetic waves. They also have practical applications in fields such as antenna design, radar technology, and wireless communications.

4. How are retarded potentials calculated for an oscillating electric dipole?

The calculation of retarded potentials for an oscillating electric dipole involves solving Maxwell's equations using mathematical techniques such as vector calculus and Fourier analysis. The resulting equations can then be used to determine the electric and magnetic fields at any point in space and time.

5. Are there any limitations to using retarded potentials for oscillating electric dipoles?

While retarded potentials provide a useful framework for understanding the behavior of oscillating electric dipoles, they do have some limitations. For example, they assume that the dipole is small compared to the wavelength of the electromagnetic radiation, and they do not take into account quantum effects. Additionally, they may not accurately describe the behavior of highly complex or nonlinear systems.

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