# Why a charged sphere whose radius oscillates in and out won't radiate?

• JD_PM
In summary, Griffiths asserts in chapter 10 of Introduction to Electrodynamics that a charged sphere with oscillating radius does not radiate due to perfect spherical symmetry and the fact that the field is a spin-1 field, with the monopole term being necessarily static. This means that for EM-waves to be propagating radially outwards, there needs to be at least a dipole component. In contrast, for gravity, which is a spin-2 field, the quadrupole contribution is the minimum required for radiative solutions.
JD_PM
In chapter 10 (Radiation; just after example 2 from 'Radiation from an arbitrary source') of Introduction to Electrodynamics by G. Griffiths he asserts that a charged sphere with oscillating radius doesn't radiate because, by Gauss law, ##E## stays the same no matter where the charges are located (either around a inner surface enclosed by a Gaussian sphere or the center of the Gaussian sphere):

$$\vec E = \frac{kQ}{r^2}\hat {r}$$

But I don't get it. If the sphere's radius is oscillating the picture I have in my mind is the charged ball kind of bouncing back and forth, which would mean that the sphere is being accelerated and thus it should radiate...

Why is Griffiths saying it will not radiate?

I think he is considering this case as that of an electric monopole, however that one shouldn't oscillate.

Thanks.

JD_PM said:
the picture I have in my mind is the charged ball kind of bouncing back and forth
Think instead of a balloon being inflated and deflated

JD_PM and Ibix
Ohh so the sphere is just expanding/contracting but its centre of mass is indeed at the same point all the time, i.e we can regard all the time Q as being at the centre.

Dale
JD_PM said:
Ohh so the sphere is just expanding/contracting but its centre of mass is indeed at the same point all the time, i.e we can regard all the time Q as being at the centre.
Note that EM-waves are transverse waves. In which tangential direction would the E-field vary given perfect spherical symmetry?

A.T. said:
In which tangential direction would the E-field vary given perfect spherical symmetry?

##\vec E## field of a point charge goes radially outwards/inwards (depending upon the charge; positive outwards and negative inwards by definition).

If you enclose the charge within a Gaussian surface, the ##\vec E## field will always be equal to (no matter how large the radius is):

$$\vec E = \frac{kQ}{r^2}\hat {r}$$

JD_PM said:
##\vec E## field of a point charge ...
That wasn't the point. In an EM-wave E oscillates perpendicularly to the propagation direction. But due to spherical symmetry in this case there is no preferred transverse direction it would oscillate about. From this one can intuitively see that no EM-waves are propagating radially outwards.

JD_PM
The reason is that the electromagnetic radiation field is a spin-1 field and thus the partial-wave expansion starts with the dipole term. The monopole term is necessarily static, i.e., the only spherically symmetric solution of Maxwell's equations is the electrostatic Coulomb field. You need at least a dipole component to get radiative solutions.

For gravity (linearized general relativity) it's even starting with the quadrupole contribution, which is because the corresponding field is a spin-2 field.

weirdoguy

## 1. Why doesn't a charged sphere whose radius oscillates in and out radiate?

The reason why a charged sphere whose radius oscillates in and out does not radiate is because it does not experience acceleration. According to Maxwell's equations, accelerating charges are the source of electromagnetic radiation. Since the radius of the sphere is simply changing back and forth, there is no net acceleration and therefore no radiation.

## 2. Does this mean that any oscillating charged object won't radiate?

No, not necessarily. The key factor is whether or not the object experiences acceleration. If the oscillations of the charged object result in a net acceleration, then it will radiate electromagnetic waves. However, if the oscillations do not result in a net acceleration, then no radiation will be produced.

## 3. How does this relate to the concept of a dipole antenna?

A dipole antenna is an example of an oscillating charged object that does radiate. This is because the two ends of the antenna have opposite charges and as the antenna oscillates, there is a net acceleration of the charges. This results in the emission of electromagnetic waves.

## 4. Can an oscillating charged sphere ever radiate?

Yes, an oscillating charged sphere can radiate if the oscillations are not perfectly symmetrical. For example, if the oscillations are not purely radial and have some angular component, then there may be a net acceleration and radiation can occur.

## 5. Is there any way to make a charged sphere with oscillating radius radiate?

Yes, one way to make a charged sphere with oscillating radius radiate is to introduce an external force that causes the oscillations. This external force can result in a net acceleration of the charges and thus, radiation can be produced. Another way is to introduce an asymmetry in the oscillations, as mentioned in the previous answer.

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