Is the electromagnetic mass real? In his Lectures on Physics vol II Ch.28-2 Feynman calculates the field momentum of a moving charged sphere with charge $q$, radius $a$ and velocity $\mathbf{v}$. He finds that the total momentum in the electromagnetic field around the charged sphere is given by:
$$\mathbf{p} = \frac{2}{3} \frac{q^2}{4\pi \epsilon_0} \frac{\mathbf{v}}{ac^2}.$$
He calls the coefficient between the field momentum, $\mathbf{p}$, and the velocity, $\mathbf{v}$, the electromagnetic mass:
$$m_\textrm{elec}=\frac{2}{3} \frac{q^2}{4\pi \epsilon_0 a c^2}.$$
He claims that this electromagnetic mass $m_\textrm{elec}$ has to be added to the standard "mechanical mass" of the sphere to give the total observed mass of the object.
Would this view be accepted by most physicists today?
Have any experiments been performed that show the effect of the additional electromagnetic mass on the dynamics of a macroscopic charged object?
I guess the problem is that such an effect would only be large enough to be observable for charged particles like electrons. In that case it would be difficult to distinguish mechanical mass, presumably due to the Higgs field, from electromagnetic mass. Maybe one could perform a high energy/short length scale experiment on an electron that excluded the effect of the electromagnetic mass? 
 A: 
this electromagnetic mass $m_\textrm{elec}$ has to be added to the standard "mechanical mass" of the sphere to give the total observed mass of the object.


 Would this view be accepted by most physicists today?

That is a wrong idea. Why?
There is no reason (in this case) to count quantity $m_\textrm{elec}$, defined based on EM momentum distributed in the whole infinite space, to total mass of the sphere.
Since $m_\textrm{elec}$ is introduced in a setting where the sphere moves, let's think about this setting. Presumably the ideas behind counting $m_\textrm{elec}$ as contribution to inertial mass are something like these:

*

*when an external (non-electromagnetic) force $\mathbf F_\textrm{ext}$ acts on the charged sphere, the equation of motion of the sphere can be written as
$$
\frac{d}{dt} \bigg( \gamma m_\textrm{mech}\mathbf v \bigg)=\mathbf F_\textrm{ext} + \mathbf F_\textrm{em.stress}
$$
where $m_\textrm{mech}$ is mechanical mass of the uncharged sphere and $\mathbf F_\textrm{em.stress}$ is EM self-force acting on the sphere due to its own charges and expressible by the Maxwell stress tensor.


*The em. stress force can be expressed as
$$
\mathbf F_\textrm{em.stress} = -\frac{d\mathbf P_\textrm{Poynting}}{dt}
$$
where $\mathbf P_\textrm{Poynting}$ is Poynting momentum (integral of the Poynting momentum density) of the field outside the sphere.


*$\mathbf P_\textrm{Poynting}$ is, at least for low velocities, directly proportional to velocity of the sphere, similarly to mechanical momentum:
$$
\mathbf P_\textrm{Poynting} = m_\textrm{em} \mathbf v
$$
where $m_\textrm{em}$ is coefficient dependent on charge and size of the sphere. It was named electromagnetic mass.
Together, these 3 ideas lead to conclusion that charged sphere has greater mass.
Idea 1. is true in macroscopic theory; it is a formulation of equation of motion in the presence of EM and non-EM forces.
The idea 2. is true only provided flux of EM momentum to or from the infinity is zero. This is true if the field is electrostatic outside some imaginary sphere containing the material sphere in question. Granted, this may be assumed true since we have no evidence to the contrary - who knows what's out there. Still, it is a special assumption with no ground in experience.
(EM momentum outside the sphere has no immediate link to the motion of the sphere. If you're thinking that rate of change of $\mathbf P_\textrm{Poynting}$ gives minus the rate of change of the remaining momentum in the sphere and its inside, this is unjustified because we do not know that total momentum of the world is constant - that depends on the state of the field at infinity and that is not part of the theory.)
The idea 3. is valid only as long as the sphere moves rectilinearly. For such motion, the inertial mass of the sphere does not manifest in any way. The sphere needs to be subjected to external force and change in its velocity needs to be measured. When the velocity changes, the EM field in the observer frame is no longer that of rectilinerly moving sphere, but contains ripples. This means the formula in 3 is no longer justified and in most cases it will be invalid.
This means the idea of EM mass based on EM momentum of field outside is erroneous. It is also unnecessary, since we have the equation of motion (see 1.) that can be used to model the motion of the sphere (with normal mechanical mass only).
This does not mean there is no electromagnetic mass though; only that in this particular case, it was calculated in an invalid way.
Systems of charges that remain together for some time and have non-zero EM energy distributed in their vicinity do experience mass defect (positive or negative) due to mutual EM interaction. For example, positive and negative particle close together will have positive mass defect - lower inertial mass than is sum of their mechanical masses.

Have any experiments been performed that show the effect of the additional electromagnetic mass on the dynamics of a macroscopic charged object?

I do not think so.
A: This view would not be accepted by physicists today.
Charged particles have mechanical mass, momentum, and energy (rest and kinetic) and the fields have energy and momentum. Total energy is conserved. Total momentum is conserved.
Are there cases where it can be sensible to imagine field momentum as an additional mechanical momentum? Sure, consider the paper "Electrostatic potential energy leading to an inertial mass change for a system of two point charges" by Timothy Boyer in the American Journal of Physics 46(4) 383-385 (1978); http://dx.doi.org/10.1119/1.11328
It's a short paper but the point is that if you ignore the forces that the charges exert on each other then they can together and collectively act like a particle of different mass.  In reality there is more than one particle, each with their own mass, their own mechanical energy, and their own mechanical momentum. And there are fields, both external and from each charge. And the fields collectively have field energy and field momentum. And when you exert forces on the charges each particle feels a force and changes its energy and momentum accordingly and they also exchange energy and momentum with the fields through which the charged particles within the system also affect each other.
So it's not that you must add field momentum to bare mechanical momentum to get some kind of total mechanical momentum. The correct physics is that you need total momentum which includes all the mechanical momentum (i.e. $\gamma m \vec v$ for each particle of mass $m$) and all the field momentum. And the only deviation allowed is that if you want to ignore some effects you can try to get away with doing it wrong by trying to compensate by adjusting some other things.
But be warned. Sometimes people fudge things in a frame dependant manner. For instance with your charged sphere you have to include the binding energy keeping the charge on the sphere before you get something that is relativistically covariant. If you include everything then it works out fine. But if you've included everything you just have the regular mechanical momentum of each charge and the total field momentum from the total field. Or more likely, you measure changes in momentum.
Also, it can be important to have momentum located in the correct place for relativistic reasons.
A: As you can figure out from the textbook, this "field momentum" moves according to the Maxwell equations where the electron is a regular point-like charge. And the charge moves according to its mechanical equation with a phenomenological (experimental) mass. So, there is no need to add anything to the latter - everything is already OK. When one adds the electromagnetic mass to the experimental mass, one introduces an error in the equation. The solutions become bad. To fix this error, one adds also a "bare mass" of the opposite sign to cancel the electromagnetic mass. Thus, nothing is left from it in the equations. I can safely say that there is no electromagnetic mass. But there is an electromagnetic mass defect due to interaction (not due to "self-action").
A: "Would this view be accepted by most physicists today?"
No it would not! 
Why not? Because Thomson in 1881, and therefore also Feynman and all of QM who use Thomson's derivation, violate the energy conservation law.  
Thomson, Feynman and QM ignore unjustly, with their derivation in Lectures on Physics vol II Ch.28-2, 1/3 of the electromagnetic mass.
The violation of the energy conservation law by Thomson, Feynman and QM in general is demonstrated in section 4 "The Electromagnetic Mass" of the article "The Equivalence of Magnetic and Kinetic Energy" .
This mistake is also the origin of the famous 4/3 problem ikn Physics!
