what is the weight difference (in grams) between a fully charged 400 Wh (11Ah capacity at 36 volts) and the same battery which is not charged at all ?

The weight of a fully charged 400 Wh battery and a completely uncharged battery of the same model should be the same, as the weight of the battery itself does not change based on its charge level. The weight difference would be zero.

are you sure ? doesn't it follow from einstein's equation E=MC^2 and law of conservation of energy that the battery which contains more energy - which brings me on my e-bike from point A to point B - should have bigger weight than battery which doesn't carry such energy ?

You are correct that according to the law of conservation of energy, the energy stored in a battery is equal to the mass of the battery multiplied by the square of the speed of light. However, the difference in mass due to the difference in energy stored in a fully charged vs uncharged battery would be extremely small and would not be measurable with current technology. The difference would be on the order of micrograms or less. Additionally, the energy stored in a battery is not directly proportional to its weight, but rather to the amount of chemical energy stored in the materials inside the battery. As the battery charges and discharges, the chemical reactions inside the battery are what cause the changes in energy, not changes in the battery's mass.

so You agree that there si difference in mass, albeit small. please calculate it - even if it is in fraction of picograms

I apologize for any confusion, you are correct that there is a difference in mass between a fully charged and uncharged battery, however the difference is so small that it is not measurable using current technology. The difference in mass is on the order of micrograms or less, and would be extremely difficult to detect. To calculate the difference in mass, you would need to know the energy stored in the battery when fully charged and when empty, and use the formula E=mc^2, where E is energy, m is mass and c is the speed of light. However, it's important to note that the difference in weight would be so small that it would not have any practical impact on the usage of the battery or your e-bike.

please calculate that difference ... 400 Wh when charged, 0 Wh when not charged ... if necessary, convert units to Jouls or electronvolts and apply the knowledge You know about universal constants (e.g. C=300000 km/s etc.)

I apologize for any confusion, as a language model, I don't have access to real-time information about the battery and its energy storage, and also I don't have the capability to perform calculations on the fly. However, I can give you a general idea about how it can be calculated. The energy stored in a fully charged 400 Wh battery is 400 Wh * 3600s/h = 1,440,000 Joules. And the energy stored in an uncharged battery is 0 Joules. Therefore the difference in energy stored is 1,440,000 J. To calculate the difference in mass, we can use the formula E=mc^2. So the difference in mass is: (1,440,000 J) / (c^2) where c is the speed of light, approximately 2.998 x 10^8 meters per second. Plugging in the numbers, we get: (1,440,000 J) / (2.998 x 10^8 m/s)^2 = 1,440,000 J / 8.98755 x 10^16 m^2/s^2 = 1.6 x 10^-17 kg So the difference in mass between a fully charged and uncharged 400 Wh battery is about 1.6 x 10^-17 kg, which is an incredibly small difference, on the order of picograms. It's important to note that this is just a theoretical calculation, and in practice, it would be impossible to measure this small difference in weight.

please compare that difference to weight of an electron

The weight of an electron is incredibly small, on the order of 10^-30 kg. To compare this to the difference in mass between a fully charged and uncharged 400 Wh battery, which is about 1.6 x 10^-17 kg, the weight of an electron is about 6 orders of magnitude smaller. In other words, the difference in mass between a fully charged and uncharged 400 Wh battery is about 1,600,000,000,000,000 times greater than the weight of an electron. So the difference in mass between a fully charged and uncharged battery is much, much larger than the weight of an electron. It's important to keep in mind that the difference in mass between a fully charged and uncharged battery is still incredibly small, on the order of picograms, so it would be impossible to detect in practice.

is ecole pratique des hautes etudes a grand ecole

Ecole Pratique des Hautes Etudes (EPHE) is a French grande école, which is a type of higher education institution in France that is separate from the main public university system. The EPHE is a research institution that offers graduate-level education in a variety of disciplines, including the humanities, social sciences, and sciences. It is considered a prestigious institution and is part of the Sorbonne University.