Question

Define each symbol in Einstein's equation: \(\mathrm{E}=\mathrm{mc}^{2}\).

Short answer

In Einstein's equation: \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum.

Step-by-step solution

Understand the Equation

Einstein's equation, \[E=mc^2\] is one of the most famous equations in physics, representing the relationship between energy and mass.

Define Symbol 'E'

The symbol \( E \) in the equation stands for 'energy.' It represents the total energy of an object, measured in joules (J). Energy is the capacity of a physical system to do work.

Define Symbol 'm'

The symbol \( m \) represents 'mass.' It refers to the amount of matter in an object, measured in kilograms (kg) in the International System of Units (SI). Mass is a measure of an object's resistance to acceleration when a force is applied.

Define Symbol 'c'

The symbol \( c \) stands for the 'speed of light in a vacuum.' It is a constant, approximately equal to \( 3.00 \times 10^8 \) meters per second (m/s). The speed of light is a fundamental constant of nature that represents the maximum speed at which information or matter can travel.

Interpretation of the Equation

The equation describes how mass (\( m \)) can be converted into energy (\( E \)) and vice versa, with the speed of light (\( c \)) as the conversion factor. It illustrates the principle of mass-energy equivalence, indicating that a small amount of mass can be converted into a large amount of energy.

Key concepts

Energy

In the realm of physics, energy is a pivotal concept linked to both motion and potential. It denotes the capacity of a system to perform work - be it lifting, heating, or even moving an object.

There are various forms of energy, each with its own distinct characteristics:

• Kinetic energy: Relates to an object's motion. For example, a rolling ball possesses kinetic energy.
• Potential energy: Tied to position or state. A book on a shelf harbors potential energy due to its height above the ground.
• Thermal energy: Pertains to the energy contained within a system due to the motion of its particles, often felt as heat.

In Einstein's equation, energy (\(E\)) is depicted as an extensive quantity, usually expressed in joules (J). It highlights the remarkable capability of matter to release vast amounts of energy, underpinning phenomena such as nuclear reactions.

Mass

Mass is often synonymous with the "amount of stuff" in an object. It is a fundamental property representing how much matter it contains.

In the context of physics, mass is essential to understand because:

• It remains constant regardless of location. Whether on Earth, the Moon, or Mars, an object's mass is unchanged.
• It contributes to the concept of inertia, which is an object's resistance to change in its motion. More mass often means more resistance to movement.

The unit of mass in the International System of Units (SI) is the kilogram (kg). In Einstein's equation, mass (\(m\)) is transformed into energy, exhibiting how matter can undergo significant changes under certain conditions.

Speed of Light in a Vacuum

The speed of light in a vacuum, symbolized by \(c\) in Einstein's equation, is a cornerstone of modern physics. It is the ultimate speed limit of the universe.

Some fascinating facts about the speed of light include:

• It has a fixed value of approximately \(3.00 \times 10^8\) meters per second (m/s). This consistency forms the basis for various measurements in scientific research and technology.
• It determines how quickly light travels from one point to another, impacting everything from solar observations to satellite communications.

In the context of mass-energy equivalence, the speed of light acts as a powerful conversion factor, reflecting the vast multiplying effect it has when mass is transformed into energy.

Mass-Energy Equivalence

The concept of mass-energy equivalence revolutionized our understanding of physics. It suggests that mass and energy are interchangeable, serving as two forms of the same essence.

This principle, encapsulated in the equation \(E=mc^2\), emerged from Albert Einstein's theory of relativity:

• It demonstrates that an object's mass can be transformed into a huge amount of energy, and conversely, energy can manifest as mass.
• It offers a foundation for modern physics realms, including nuclear power generation and the energy produced in stars.

The mass-energy equivalence underscores the profound interconnectedness of physical properties, revealing how the universe conserves energy by transforming mass in extraordinary ways.