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Imagine you had a magical formula that could describe how the mass of an object could be transformed into a vast amount of energy. Well, we don't need to rely on magic, thanks to Albert Einstein. He provided us with one of the most famous equations in physics: E = mc^2. This principle of mass-energy equivalence is a cornerstone in modern physics and has profound implications.

Simply put, the equation means that energy (E) equals the mass (m) of an object times the speed of light squared (c^2). Given that the speed of light is an immensely large number (approximately 299,792,458 meters per second), even a tiny amount of mass can be equivalent to a huge amount of energy. That's why, in the textbook exercise above, a mere 1-gram raisin could theoretically power a large city for hours! This astonishing conversion shows how potent the relationship between mass and energy is.

While it's a spectacular thought, the conversion of mass to energy is not just theoretical; it has practical real-world applications. For instance, nuclear reactors work on the principle that a small amount of mass is converted into a significant amount of energy, releasing energy harnessed as electricity.

Practical Example

Using the same formula, E = mc^2, we can see how something as small as a raisin, with a mass of 1 gram, could yield \(9 \times 10^{13}\) joules of energy. This stunning amount of energy comes about because of the aforementioned high value of the constant c (the speed of light). However, in reality, transforming 100% of mass into energy is a feat not yet achievable by our current technology outside of particle-antiparticle annihilation scenarios. But the equation allows us to make theoretical calculations that serve as powerful illustrations and guide our scientific endeavors.

Understanding power consumption and being able to calculate it effectively is crucial in various fields of engineering and energy management. Power is the rate at which energy is used, typically measured in watts (W) or joules per second (J/s). A watt equates to one joule of energy consumed per second. Therefore, when we talk about a 1 gigawatt (GW) city, we're saying it uses \(1 \times 10^{9}\) joules of energy every second.

To calculate the duration that a given energy source can supply this city, we divide the total energy by this rate. For the exercise’s 1-gram raisin and its equivalent energy, it would last for \(9 \times 10^{4}\) seconds or approximately 25 hours. This piece of information can be particularly enlightening when comparing different energy sources or planning for energy needs in various scenarios.