Question

Energy equivalent of \(0.001 \mathrm{mg}\) is (a) \(9 \times 10^{7} \mathrm{ergs}\) (b) \(9 \times 10^{9} \mathrm{ergs}\) (c) \(9 \times 10^{7} \mathrm{~J}\) (d) \(9 \times 10^{5} \mathrm{~J}\)

Short answer

The energy equivalent of \(0.001 \text{ mg}\) is (c) \(9 \times 10^{7} \text{ J}\).

Step-by-step solution

Understand the Problem

We need to find out how much energy is equivalent to a mass of \(0.001 \text{ mg}\) using the equation \(E = mc^2\). The energy will be calculated in ergs and then we'll check it against the given options.

Convert Mass to Kilograms

First, we need to convert the mass from milligrams to kilograms because the SI unit of mass is the kilogram. Since \(1 \text{ mg} = 10^{-6} \text{ kg}\), \(0.001 \text{ mg} = 0.001 \times 10^{-6} \text{ kg} = 10^{-9} \text{ kg}\).

Calculate Energy using E=mc^2

Now, we use the equation \(E = mc^2\) to find the energy. Here, \(m = 10^{-9} \text{ kg}\) and \(c = 3 \times 10^8 \text{ m/s}\). Thus, \[E = (10^{-9} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 = 9 \times 10^{-9} \times 10^{16} \text{ J} = 9 \times 10^7 \text{ J}\].

Convert Joules to Ergs

We now convert the energy from joules to ergs. Recall that \(1 \text{ J} = 10^7 \text{ ergs}\). Therefore, \(9 \times 10^7 \text{ J} = 9 \times 10^7 \times 10^7 \text{ ergs} = 9 \times 10^{14} \text{ ergs}\).

Verify the Correct Option

The calculated energy equivalent is \(9 \times 10^{14} \text{ ergs}\), which does not match the given options because the conversion to ergs was incorrect. Re-evaluate the conversion step.

Correct Conversion and Final Answer

Let's correct this: The energy in joules is \(9 \times 10^7 \text{ J}\), and each joule is \(10^7\) ergs, retaining the same value in order of magnitude for the options. Re-evaluating options based on energy in joules shows that the answer matches the energy in joules: (c) \(9 \times 10^7 \text{ J}\).

Key concepts

Mass-Energy Equivalence

The concept of mass-energy equivalence is one of the most profound insights from the realm of physics. Albert Einstein introduced this relationship through his remarkable equation, \( E = mc^2 \). Here's what it means at its core:

• Mass and Energy as Interchangeable: The equation signifies that mass can be converted into energy and vice-versa. It establishes a direct relationship where a small amount of mass can potentially become a vast amount of energy. This principle is crucial in fields like nuclear physics.
• The Equation: The formula consists of three components: \( E \) is energy, \( m \) represents mass, and \( c \) stands for the speed of light in a vacuum, approximately \( 3 \times 10^8 \ \text{m/s} \).
• Implications: Even a tiny mass can be equivalent to a large energy amount due to the speed of light being squared, showing why nuclear reactions release massive energy.

Understanding mass-energy equivalence allows you to comprehend the sheer power held within even the smallest particles.

Unit Conversion

Unit conversion is the process of changing a measure from one unit to another. It is crucial not only in solving textbook problems but also in practical applications across science and engineering. Here's why:

• Consistency of Units: Having measurements in consistent units ensures that calculations remain correct and meaningful. When dealing with equations like \( E = mc^2 \), using proper SI units is necessary.
• Standard Units: The international system of units (SI) is universally accepted for scientific calculations, and it includes units like kilograms for mass and meters for length.
• Conversion Process: When converting measurements, use multiplication or division to switch from one unit scale to another. For example, converting milligrams to kilograms involves multiplying by \( 10^{-6} \) since \( 1 \text{ mg} = 10^{-6} \text{ kg} \).

By understanding and applying unit conversion, you can solve problems with varying units effortlessly, ensuring accuracy and coherence in scientific work.

Joules to Ergs Conversion

To convert energy from joules to ergs, understanding the relationship between these two units is essential. Joules and ergs are units of energy measurement, but they differ in magnitude:

• The Relationship: One joule is equivalent to \( 10^7 \) ergs. This means the joule is a much larger unit than the erg.
• Conversion Process: To convert from joules to ergs, you multiply the number of joules by \( 10^7 \). For example, if you have \( 9 \times 10^7 \) joules, to convert this to ergs, you do the following calculation: \[ 9 \times 10^7 \ \text{J} \times 10^7 = 9 \times 10^{14} \ \text{ergs} \] However, always verify that this aligns with the problem requirements.
• Practical Application: This unit conversion may appear in fields like astrophysics or when studying microscopic systems where energy levels might be more conveniently expressed in ergs.

Mastering the conversion between joules and ergs enhances your ability to work with different energy measures and understand the scale of energy more precisely.